Embed Size (px)
Citation preview
Unit 2 – Congruence
Unit 2 – Congruence WARM-UPS
Unit 2 Notesheet(Congruenc
e)
Day 1 – Transformatio
n wkst
Day 2 – Congruent
Triangles #1 wkst
* Congruent Triangles #2
wkst
More w/Congruent
Triangles wkst
Quiz Review wkst
CPCTC Proofs
Triangle Theorems
wkst
Midsegments wkst
Parallelogram Practice wkst
Parallelogram Proof Practice
wkst
Complete Review wkst
Quick Check Quick Check
Daily Work Score ________
Quotes Written on Back? Yes No
________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Unit 2 – Congruence and Similarity
Day 1* Notes on Transformations* Class Practice
* Day 1 – Transformation wkst
Day 2* Transformation Warm-Up/Challenge* CPCTC and Triangle Congruence Notes (SAS, SSS, ASA, AAS, HL)
* Day 2 – Congruent Triangles #1 wkst
Day 3* Quick Check #1 – Triangle Congruence* Warm – Up – Congruent Triangles #2 wkst* Congruent Triangles Proofs
* Congruent Triangles #2 wkst
Day 4 * Warm – Up (Reflection Rules)* Group Work on More w/Congruent Triangles wkst* CPCTC Proofs examples* Quick Check #2 - Congruent Triangles
* More w/Congruent Triangles wkst* CPCTC Proofs wkst
Day 5* Review for Quiz * Quiz Review wkst
Day 6* Triangle Congruence Quiz* Begin notes on Triangle Theorems (acute, right, scalene, isosceles, etc.)
Day 7 * Finish up notes on Triangle Theorems* Midsegment notes
* Triangle Theorems wkst* Midsegments wkst
Day 8* Parallelogram Notes * Parallelogram Practice wkst
Day 9* More with Parallelograms* Kites and Trapezoids Notes/Practice
* Parallelogram Proof Practice wkst* Finish up Kites/Traps Practice
Day 10 * Review for test* Complete Review wkst
Day 11 TEST DAY
Day 1 – Notes on Congruence TransformationsWords/Terms You NEED to know:
ANGLE OF ROTATION the measure of the angle created by the P_________ vertex to the point
of R_________ to the image vertex. All of these angles are C__________
when a figure is rotated.
CONGRUENCY TRANSFORMATION a transformation in which a G__________ F_________ moves but
keeps the same S______ and S_________
T______________ R______________ R_________________
CORRESPONDING ANGLES angles of two figures that lie in the S______P___________ relative to
the figure. In transformations, the corresponding vertices are the
P__________ and the I__________ vertices, so ∡ A∧∡ A ' are corresponding vertices and so on.
CORRESPONDING SIDES sides of two figures that lie in the same P__________ relative to the
figure. In transformations, the corresponding sides are the P_________
and I________ sides, so AB and A′B′ are corresponding sides and so on.
EQUIDISTANT the same distance from a R______________ P_________.
IMAGE the N______, resulting figure A_________ a transformation
(the image is the figure the arrow is pointing to)
(the preimage is the figure where the arrow starts)
PREIMAGE the O__________ figure B___________ undergoing a transformation
ISOMETRY a transformation in which the preimage and image are C____________
RIGID MOTION a transformation done to a figure that M______________ the figure’s
S_______ and S___________ or its segment lengths and angle measures. This motion results in an I______________.
LINE OF REFLECTION the P_______________ B___________ of the segments that connect the
corresponding vertices of the P__________ and the I_________.
POINT OF ROTATION the F__________ location that an object is turned around; the point can
lie O___, I__________, or O___________ the figure
There are 3 main types of transformations we will work with.
T_________________• A translation is sometimes called a S________________.• In a translation, the figure is moved H___________________ and/or V_____________________.• The orientation of the figure remains the S_____________. (it faces the same direction)
• Connecting the corresponding vertices of the preimage and image will result in a set of P_______________ L__________.
R____________________• A reflection creates a M_____________ I_______________ of the original figure over a reflection line.• A reflection line can pass T_________________ the figure, be ______ the figure, or be _________ the figure.• Reflections are sometimes called F____________.• The O_______________________ of the figure is changed in a reflection. (it faces a different direction)
• In a reflection, the corresponding vertices of the preimage and image are E_____________ from the line of reflection, meaning the distance from each vertex to the line of reflection is the same.• The line of reflection is the P________________ B_________________ of the segments that connect the corresponding vertices of the preimage and the image.
R_______________• A rotation moves all points of a figure along a C___________ A______ about a point. Rotations are sometimes called T___________• In a rotation, the O____________________is changed.• The point of rotation can lie _____, ______________, or __________________ the figure, and is the fixed location that the object is turned around.• The A____________ of R_______________ is the measure of the angle created by the preimage vertex to the point of rotation to the image vertex. All of these angles are congruent when a figure is rotated.• Rotating a figure C___________ moves the figure in a circular arc about the point of rotation in the same direction that the hands move on a clock.• Rotating a figure C________________ moves the figure in a circular arc about the point of rotation in the opposite direction that the hands move on a clock.
* This chart tells you how you can adjust your x and y – coordinates for each kind of transformation. Use these ideas on the practice problems that follow.
TRANSFORMATION COORDINATE MAPPING & DESCRIPTION
Translation (x, y) → (x + a, y + b) Translation ‘a’ units horizontally and ‘b’ units vertically.
Reflection (x, y) → (-x, y) Reflection across y-axis(x, y) → (x, -y) Reflection across x-axis
(x, y) → (y, x) Reflection across line y = x(x, y) → (-y, -x) Reflection across line y = -x
Rotation (x, y) → (y, -x) Rotation about (0,0), 900 clockwise(x, y) → (-y, x) Rotation about (0,0), 900
counterclockwise(x, y) → (-x, -y) Rotation about (0,0) 1800
Practice1. Starting with the figure below, move the figure according to these transformations. Label each new figure accordingly. Move from #1 to #2 to #3, etc.
2) (x + 3, y + 10).
3) (x – 8, y + 1)
4) (x, y – 4)
5) (x + 5, y)
Where does point B end up after #5? ______________
2. Starting with the given figure, follow the given REFLECTIONS. Move from figure #1 to #2 to #3,etc. Label each figure. Write out the new coordinates for point A..
2) rx-axis : A’ = (_______, _______)
3) ry-axis: A’ = (________, _______)
4) rx-axis: A’ = (________, _______)
5) ry-axis: A’ = (________, _______)
6) ry=x: A’ = (________, _______)
Where does your figure finish up after #5?
______________________________________
3. Starting with the given figure, follow the given REFLECTIONS. Move from figure #1 to #2 to #3,etc. Label each figure. Write out the new coordinates for point A..
2) r(y = -3) : A’ = (_______, _______)
3) r(x = 1) : A’ = (________, _______)
4) r(y = 2) : A’ = (________, _______)
5) r(x = -2): A’ = (________, _______)
6) ry = -x: A’ = (________, _______)
Where does your figure finish up after #5?
______________________________________
4. Starting with the given figure, follow the given ROTATIONS. Move from figure #1 to #2 to #3,etc. Label each figure. Write out the new coordinates for all points.
1) 900 around origin clockwise:A’ = (_________, ________)B’ = (_________, ________)C’ = (_________, ________)
2) 1800 around origin counterclockwise:
A’ = (__________, _________)B’ = (_________, ________)C’ = (_________, ________)
3) 900 around origin counterclockwise:
A’ = (__________, _________) B’ = (_________, ________)C’ = (_________, ________)
5. Determine whether the polygons with the given vertices are congruent. Support your answers by completing the transformation.
a) A(-2,-2); B(-4, -1); C(-1, -1); and T(2, 2); U(4, 1); V(1, 1)
Is this a translation, reflection, or
rotation? ________________________
(x, y) →(__________, ____________)
6) Prove the following polygons are congruent.
J(-5, 2); K(-2, 5); L(-2, 2) and M(5, 0); N(2, 3); O(2, 0)
Is this a translation, reflection, or
rotation? ________________________
x, y) →(__________, ____________)
Day 2 - Corresponding Parts of Congruent Triangles are Congruent( CPCTC )
To start the next part of Unit 2, we will be looking at Congruent Triangles. By saying two triangles are congruent, we mean that the t____________ a______________ in one triangle are the same
measure as the t________________ a________________ in the second triangle and the t_____________ s__________ in one triangle are the same length as the t_____________
s_______________ in the second triangle.
You can identify if two triangles are congruent by looking at pictures and using T__________ M__________
* In the given pictures, you can see that sides AB and DE both have 1 tick mark on them –
this means these two sides are congruent. The same is true for sides BC and EF (2 tick marks)
and AC and DF (3 tick marks)
** In regards to the angles, look at the arcs marked on the pictures. ∡A is the same as ∡D (1 arc); ∡B is the same as ∡E (2 arcs); and ∡C is the same as ∡F (3 arcs).*** Knowing the corresponding sides and angles, we can make a congruency statement:
Δ__________ ≅ Δ___________This is not the only way to state the triangle congruency; we could also state…
Δ__________ ≅ Δ__________ or Δ__________ ≅ Δ___________What is true in all three ways the triangle congruency is stated?
(A and D are in the same place, as are B and E, and C and F)
Use this idea to state the triangle congruencies below:
1) Δ_______ ≅ Δ________ 2) Δ_______ ≅ Δ________
Using the congruency statements you wrote above in #1 and #2, rewrite them in 2 other ways.
1) Δ_______ ≅ Δ________ 2) Δ_______ ≅ Δ________
1) Δ_______ ≅ Δ________ 2) Δ_______ ≅ Δ________
Now try to go the other direction. You will be given a congruency statement and you will need to put the tick marks and letters on the pictures.
3) ΔDAV ≅ ΔAJE 4) ΔPLE ≅ ΔKLE
Fill in the congruencies from the triangles above:
∡D ≅ ∡ _______ ∡P ≅ ∡ _______∡A ≅ ∡ _______ ∡L ≅ ∡ _______∡V ≅ ∡ _______ ∡E ≅ ∡ _______DA ≅ PL ≅ AV ≅ ¿ ≅ VD ≅ EP ≅
* There are 6 congruencies for every set of congruent triangles. In short, these 6 congruencies are stated as CPCTC or Corresponding Parts of Congruent Triangles are Congruent.
(this is one of the most important concepts in this unit – know it well!!!)
State the six congruencies for the given congruency statements.
5) ΔBUC ≅ ΔAHS 6) ΔATL ≅ ΔBRV
When triangles are congruent, the corresponding parts of the triangles are also congruent. It is also true that if the corresponding parts of two triangles are congruent,
then the triangles are congruent. It is possible to determine if triangles are congruent by measuring and comparing each angle and side, but this can take time. There is a set of
congruence criteria that lets us determine whether triangles are congruent with less information. On the next page we will look at a couple of the ways to prove two triangles
are congruent.
1) The Side-Side-Side (SSS) Congruence Statement states that if three sides of one triangle
are congruent to three sides of another triangle, then the two triangles are congruent.
ΔABC ≅ Δ DEF Δ ADT ≅ Δ EDTBoth pairs of triangles above are congruent because the three sets of sides in one
triangle are congruent to the three sides in the other triangle.
2) The Side-Angle-Side (SAS) Congruence Statement states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.
Δ EDF ≅ Δ BAC Δ ACP ≅ Δ ABP
* This congruency theorem involves using two sides of a triangle as well as the INCLUDED ANGLE. This angle must be the angle that is between the congruent sides.
3) The Angle-Side-Angle Congruence Statement, or ASA, states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.
Δ ABC ≅ Δ XYZ Δ NOM ≅ Δ POR
* This theorem involves using the INCLUDED SIDE. This is similar to the other theorem
which uses the included angle. Notice the side is in between the two congruent angles.
4) The Angle-Angle-Side (AAS) Congruence Statement, states that if two angles and a non-included
side of one triangle are congruent to the corresponding two angles and side of a second triangle, then the triangles are congruent.
Δ LKM ≅ Δ TUS Δ RPS ≅ Δ RQS
* This congruency theorem using a NON-INCLUDED side, which means the side is not in between the congruent angles.
5) The Hypotenuse – Leg Congruency Statement, states that if the Hypotenuse of one right triangle is congruent to the Hypotenuse of a second right triangle and the pair of right triangles also have a set of congruent legs, then the two triangles are congruent. This congruency statement only applies to right triangles(since only those contain a hypotenuse).
* AB and ZX are congruent hypotenuses* AC and ZY are congruent legs
ΔACB ≅ ΔZYXIn each pair of triangles, identify the theorem by which they are congruent (SSS, SAS, ASA, AAS, or HL). Then write a congruency
statement. 1. 2.
_________________ ____________________ Δ BAC ≅ Δ___________ Δ YAZ ≅ Δ___________
3. 4.
_________________ ____________________ Δ BAC ≅ Δ___________ Δ LJK ≅ Δ___________
5. 6.
_________________ ____________________ Δ QYP ≅ Δ___________ Δ ACB ≅ Δ___________
7. 8.
_________________ ____________________ Δ BPA ≅ Δ___________ Δ BAC ≅ Δ___________
Day 3 – Triangle Congruence ProofsFor each problem, complete the following: a. Show the given information in the diagram (using tick marks to show congruent sides and arcs to show congruent angles) b. Show any other congruent parts you notice (from vertical angles, sides shared in common – reflexive property, or alternate interior angles with parallel lines) c. Give the postulate or theorem that proves the triangles congruent (SSS, SAS, ASA, AAS, HL) d. Finally, fill in the blanks to complete the proof.
For the following problems, complete a two column proof from scratch. 5. Given: Y is the midpoint of XZ, AY BY and AYX BYZ.
Prove: XYA ZYB
6. Given: RTS is isosceles with legs RT and TS. Q is the midpoint of RS.
Prove: RTQ STQ
7. Given: P N and M is the midpoint of PN.
Prove:
A B
X Y Z
QR S
T
K
N
M
P
Q
Statements Reasons
Statements Reasons
8. Given: AC BD and AC BD
Prove:
Day 4 – CPCTC Proofs
A B
C
D
1 23 4
Statements Reasons
Statements Reasons
Day 6/7 - Theorems on Triangles NotesheetTRIANGLE INEQUALITY THEOREM
* A triangle is only a true triangle if the SUM of the two shorter sides is larger than the 3rd side. Examples: 5, 8, 12 7, 7, 14 9, 12, 25
_________________________ __________________________ __________________________
Types of Triangles:A__________ triangle a triangle in which all of the angles are acute (less than
90˚)R__________ triangle a triangle with one angle that measures 90˚
O__________ triangle a triangle with one angle that is obtuse (greater than 90˚)
S__________________ triangle a triangle with no congruent sidesI___________________ triangle a triangle with at least two congruent sidesE__________________ triangle a triangle with all three sides equal in length
Name each of the following triangles in two ways – by its angles and by its sides.
a) b) c)
_____________________________ __________________________ _______________________________________________________ __________________________ __________________________
The SUM of the Angles in Any Triangle is __________. (This is known as the Triangle Sum Theorem)
Exterior Angle Theorem:
The measure of an exterior angle of a triangle is equal to the S______ of the measures of its R_________ I__________
angles.
Can you explain why this is true?
PRACTICE using these theorems:
1) Find the value of ‘x’ and the measure of each angle.
2)
3) Find the measure of each angle.
4)
* Here are a couple more theorems, that we will use quite a bit during this unit (as well as in a couple years if you take trigonometry).
If one side of a triangle is L_________ than another side, then the angle opposite the L___________ side has a G______________ measure than the angle opposite the S_____________ side.
(converses)
If one angle of a triangle has a G_____________ measure than another angle, then the side opposite the G_______________ angle is L___________ than the side opposite the L___________ angle.
Using these theorems, put the angles in order Put the sides of this triangle in order from from largest to smallest. largest to smallest.
ISOSCELES TRIANGLE THEOREMS
** Make sure you know the different parts of an isosceles triangle.**
If two sides of a triangle are C___________, then the angles opposite the congruent sides are C______________.
Proof: Complete this proof of the theorem from above:
Given: CO ≅ CW CS is an altitude of ΔCOW
Prove: ∡0 ≅ ∡W
1) ________________________________________ 1) Given
2) 2) Def. of …3) ________________________________________ 3) All right angles are congruent
4) 4) __________________________________________________5) ________________________________________ 5) HL6) ∡0 ≅ ∡W 6) __________________________________________________
( The converse of this theorem is also true – If the opposite angles are congruent, then the opposite sides are also congruent.)
* Another type of triangle is an EQUILATERAL TRIANGLE – this is a triangle in which all 3 sides have the same length. What is the measure of each interior angle of this type of triangle? Why?
Use the previous Triangle Theorems to answer the following:
5) Find:
6) Find:
7) Find: m∠x = _______ m∠y = _______
8) Find x = ________ y = ________
MidsegmentsWarm UpYou will be given 6 straws that you need to create an equilateral triangle with. Do this and draw a picture of what your triangle looks like here – make sure you show all 6 straws.
You will then be given a 7th straw that you need to put inside your triangle so that it touches two sides of your triangle, but does not go past either side. It must fit perfectly. Draw a picture of what this looks like.
What is the name we give to this 7th straw? __________________________________________
-------------------------------------------------------------------------
Triangle Midsegment Theorem
A midsegment of a triangle is P__________________ to the third side and is ______________ as long.
* In the given picture, XY is a midsegment of ΔABC.
XY ⃦ AC & XY = AC½
(How can you prove these are true?)* Every triangle has _______________ midsegments. When drawn, they create a M___________________________ T_____________.
* ΔRST ~ ΔCBA
Example 1Find the lengths of BC and YZ and the measure of ∠AXZ.
Example 2If AB = 2x + 7 and YZ = 3x – 6.5, what is the length of AB?
Individual Practice:
1) Find the lengths of AC and YZ and the measure of ∠XZY.
2) If AB = 7x – 13 and YZ = 2x + 4, what is the length of YZ?
CHALLENGE: The midpoints of a triangle are X (–2, 5), Y (3, 1), and Z (4, 8). Find the coordinates of the vertices of the triangle.
Perpendicular Bisectors
A Perpendicular Bisector is a L__________, S____________, or R____________ that is P________________________ to another segment and intersects the segment at it’s M___________________.
Perpendicular Bisector Theorem: If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
If:
Then:
Converse of the Perpendicular Bisector Theorem: If a point is equidistant from the endpoints of the segment, then it is on the perpendicular bisector of the segment.
Practice: A. Find BC. B. Find AD if AC is the perpendicular bisector to BD.
C. Find TU
Day 8 – Parallelograms NotesheetProperties of Quadrilaterals
A parallelogram is a type of quadrilateral that has two pairs of opposite sides that are parallel. Parallelograms are denoted by the symbol . If a quadrilateral has two pairs of parallel, opposite sides, then it can be classified as a parallelogram.
There are 5 theorems associated with PARALLELOGRAMS:
Opposite S____________ are congruent
Opposite A____________ are congruent
C_________________ angles are
S__________________________
D________________ bisect each other
Parallelograms can be broken down into three more specific types of quadrilaterals with the same properties as parallelograms. The three specific types also have some of their own properties.
Applying Properties of Quadrilaterals
1. Solve for x, y, and z. Relationship: ____________________________________
2. Solve for x, y, and z. Relationship: ____________________________________
There are 5 theorems associated with PARALLELOGRAMS:
Opposite S____________ are congruent
Opposite A____________ are congruent
C_________________ angles are
S__________________________
D________________ bisect each other
Square
All properties of parallelograms
F____________ R_____________
A_________
F____________ C_____________
S_________
Diagonals are C________________,
P_______________, and
Rhombus
All properties of parallelograms
D_________________ are
perpendicular
Diagonals B_______________ each
other
F__________ S___________ are
congruent
Rectangles
All properties of parallelograms
D____________________ are
congruent
F__________ R____________
A____________
3. In parallelogram ABCD, AB = 17.5, DE = 18, and . Point E represents the intersection of the diagonals. Draw a picture of parallelogram ABCD and answer the following questions:
a. BD = _________ b. CD = _________
c. BE = _________ d.
e. f.
Relationship: _____________________________________
Relationship: ____________________________________
Relationship: ____________________________________
6. Find the value of y. Then find the measure of Angle C and D.
4. Find the value of x. Then find the length of BC.
5. Find the value of x. Then find Angle Q.
T U
VW
Relationship: ____________________________________
8. RSTV is a rhombus. Find the length of TV. Relationship: ____________________________________
9. In rectangle TUVW below, it is know that Relationship: ____________________________ and . Find the value of x.
Proving and Justifying with Parallelograms
7. EFGH is a parallelogram. Find w and z.
Yesterday, you explored 4 out of the 5 theorems associated with parallelograms. You learned that opposite sides are congruent, opposite angles are congruent, consecutive angles are supplementary, and diagonals bisect each other. It was mentioned that, in a parallelogram, diagonals form two congruent triangles, but you never really explored it. In the problem below, you are going to prove that a parallelogram forms two congruent triangles.
Using the picture at the right, answer the following questions about parallelogram MPRK. Justify your answer (using properties of parallelograms) for each question.
a. Why? ___________________________________
b. Why? ________________________________________________________________
c. Why? ________________________________________________________________
d. Why? ________________________________________________________________
Given: JKLM is a parallelogram
Prove:
Statements Reasons
1. JKLM is a parallelogram 1. ___________________________
2. 2. ___________________________
3. ___________________________ 3. Alternate Interior are
4. ___________________________ 4. ___________________________
5. 5. ___________________________
e. Why? ________________________________________________________________
f. Why? ________________________________________________________________
g. Why? ________________________________________________________________
h. Why? ________________________________________________________________
i. Why? __________________________________________________________
Justifying with Properties of Parallelograms
Determine if each quadrilateral must be a parallelogram. Explain why or why not. a. b. c.
d. e. f.
g. If the diagonals are perpendicular, which type of quadrilateral could it be?
h. If all four sides are the same length, which type of quadrilateral could it be?
i. If the diagonals are congruent, which type of quadrilateral could it be?
j. A parallelogram has one right angle. What is a more specific name for the parallelogram? Justify your answer using properties of parallelograms and specific quadrilaterals.
Proofs with Parallelograms
a. Given: ABCD is a parallelogram
Prove:
Statements Reasons
1. ABCD is a parallelogram 1. ______________________________
2. 2. ______________________________
3. 3. ______________________________
4. ___________________________ 4. ______________________________
b. Given: ABCD and AFGH are parallelograms
Prove:
c. Given: EFGH is a rectangle,
J is the midpoint of .
Prove: is isosceles.
Day 9 - Trapezoids and Kites
1) Here’s a picture of a couple Trapezoids. What are some properties of these figures. Use the pictures to help. __________________________________________________________________________________________________ ____________________________________________________________________________________________________________________________________________________________________________________________________
2) Name the different parts of the trapezoid.
Statements Reasons
1. ABCD is a parallelogram 1. ______________________________
2. ___________________________ 2. Given
3. 3. ______________________________
4. 4. ______________________________
5. ___________________________ 5. ______________________________
Statements Reasons
1. EFGH is a rectangle. 1. ______________________________
2. are right angles. 2. _____________________________
3. _____________________________ 3. ______________________________
4. J is the midpoint of . 4. ______________________________
5. ___________________________ 5. ______________________________
6. EFGH is also a parallelogram 6. ______________________________
7. 7. ______________________________
8. 8. ______________________________
9. ___________________________ 9. ______________________________
10. ___________________________ 10. _____________________________
Bases ________________ Legs _______________ Opposite Angles ___________________
Diagonals _______________ Opposite Sides ______________ Base Angles _______________________
3) What is the name of the segment inside a trapezoid that connects the midpoints of the legs?
________________________________
4) Explain how you find the length of this segment. _________________________________________________ _______________________________________________________________________________________________
5) Use this method to find the length of segment RT in each picture. Write out your steps.
a) b) c)
x = ___________________ x = _________________ x = __________________
6) Use the properties of trapezoids to find the indicated angles.
a) b) c)
∡A = _______, ∡D = _______ ∡A = _________ ∡C = __________ ∡D = ___________ ∡C = _________7) Here’s a picture of a Kite. Mark your figure with tick marks according to the definition of a kite. Now, draw in the long diagonal, KT. Two triangles are created. Are they congruent? ______________ By what method? ______________________Write out the congruency: ∆ KIT ≅ ∆¿ When two triangles are congruent we know that their…C_______________________ P_______________ are also congruent.
Write out the congruencies for the different angles of the two triangles.∡_________≅∡________ ∡________ ≅ ∡________ ∡________ ≅ ∡__________What properties of a kite do we learn from this?___________________________________________________________________________________________________________________________________________________________________________________________________________________
8) Now draw in both diagonals. This seperates the kite into 4 triangles. Label the intersection point of the diagonals as “P.” Look at triangles KPE and KPI. How can we show these two triangles are congruent? ________________________________________________________________ ________________________________________________________________
Which method would be used ? ________________________
With these two triangles being congruent, write out the congruency.
∆ KPE ≅∆ ¿
Write out the corresponding angles for these congruent triangles.
∡_________≅∡________ ∡________ ≅ ∡________ ∡________ ≅ ∡__________ Look at the angles that are formed at point “P.” What kind of angles do the diagonals create? _______________________ How do you know this about the angles? __________________________________________________________________________________ What property about the diagonals of kites does this tell us? ___________________________________________________________________________________________________________________9) Find the indicated parts of each kite.
a) b) c)
∡H = ________ ∡A = ________ ∡A = ________ ∡T = ________ AH = ________ MT = ________ AT = _________ MH = ________10) Fill in the table for trapezoids and kites.Property Trapezoid Isosceles Trapezoid Kite
Opposite sides are parallelOpposite sides are conguentOpposite Angles are congruentA diagonal bisects a pair of anglesDiagonals bisect each otherDiagonals are perpendicularConsecutive Angles are congruentUse the properties from above to complete the following:
11) 12)13) 614)15)16)17) In an isosceles trapezoid, if one pair of base angles is twice the measure of the second pair of base angles, what are the measures of the angles?
__________________
18) If the midsegment of a trapezoid measures 6 units long, what is true about the lengths of the bases of the trapezoid?
Complete the statement with always, sometimes or never.
19) A trapezoid is ______________________a parallelogram.
20) The bases of a trapezoid are _______________________ parallel.21) The base angles of an isosceles trapezoid are ______________________ congruent.22) The legs of a trapezoid are _______________________ congruent.
Find the value of ‘x’ and/or ‘y’, then write in the length of each side.
23) 24)
X = _______________ X = __________________
25)
