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Summer 2017 Regents Geometry
Assignment
Unit 1: Geometry Foundations
Name: __________________________________
Please read the following links on basic geometric symbols, labeling, and notation before
completing unit 1.
Page 1
Lesson 1.1: Understanding Points, Lines and Planes Point:
● represented as a ___________ ● Named by a ___________ ____________ ● No ___________, _____________, or ________ ● Indicates ____________ or _____________
Line: Set of points
● A __________ path with no ________________ or _________ ● Use any two points on a line to name it ● Sometimes labeled by a lowercase letter
Plane:
● A __________ surface with no _____________ or __________ ● Named by 3 points not on a line ● Sometimes labeled by a script capital letter
Collinear:
Set of points, all of which lie on the ______________ line
3 collinear points are: ______________ 3 non-collinear points are: ______________
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Coplanar: Set of points, all of which lie in the __________ ___________
3 coplanar points are: ______________ 3 non-coplanar points are: ______________
Line Segment:
● A part of a ________ consisting of two __________ and all points on the line between them.
● Denoted by ____ or ____
● Length or measure of a segment is the distance between ___ __________
Length of _____ is denoted by _____ Congruent Segments: Segments that have __________ ___________ Ray:
● A part of a line that consists of an __________ and all of the _________________________________________________
How many rays can you name in the figure below? Opposite Rays:
● Two rays of the same ________ with a ___________ ______________
but no other _________ in common A point on a line creates two opposite rays.
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Name the following based on the diagram below:
Line AB = _____
Line Segment AB = _____
Ray AB = _____
Length of Line Segment AB = _____
Name a pair of opposite rays _________________
Examples: 1) Draw and label a segment with endpoints U and V. 2) Draw and label opposite rays with a common endpoint Q. 3) Name the plane to the right using three non-collinear points. 4) Sketch a figure that shows two lines intersect in one point in a plane, but only one of the lines lies in the plane.
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Using the diagram below, name each of the following: 5) Two lines _____________ 6) Two planes ____________ 7) Point on BD ____________ 8) What is the intersection of the two planes? ______
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Lesson 1.2: Congruent Segments and Angles Finding the length of a segment: the distance between any two points is the ___________________________ of the difference of the coordinates.
_____ _____ _____ Congruent segments are segments that have the same length. In the diagram below, PQ = RS, so you can write: ≅ RS This is read as “segment PQ is congruent to segment RS.” Tick marks are used in a figure to show congruent segments. Midpoint: The midpoint M of is the point that bisects, or divides the segmentAB into two congruent segments.
● If M is the midpoint of AB, then AM = MB.
Example 1: If AB = 6, then AM = _____. MB = _____.
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Example 2: What letter is the midpoint of ? _____AC Example 3: What letter is the midpoint of ? _____BF Example 4: What is the coordinate of the midpoint of ? _____BE Example 5: D is the midpoint of EF, ED = 4x + 6, and DF = 7x – 9.
Find ED _____
DF _____
EF _____
A Segment Bisector is any _____, ___________, or ________ that intersects a segment at its ________. It divides the segment into two equal parts at its midpoint. Draw a bisector for this segment:
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In order for you to say that a point B is between two points A and C, all three points must lie on the same line, and AB + BC = AC. Angle: set of points that is the union of _________ ____________ having the
same ____________________.
The set of all points between the sides of the angle is the interior of an angle. The exterior of an angle is the set of all points outside the angle.
➔ When you have several angles with the same______________________,
we must use _______ letters (or one indicated number) to name an angle.
We could call the angle to the right either:
or or Right Angle: an angle whose degree measure is _____________ Acute Angle: an angle whose degree measure is ___________________________
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Obtuse Angle: an angle whose degree measure is ___________________________
________________________________________________________________ Straight Angle: an angle whose degree measure is formed by two______________ _______ and measures ______. Congruent angles are angles that have the same measure. In the diagram, m∠ABC = m∠DEF, so you can write ∠ABC ≅ ∠DEF. This is read as “angle ABC is congruent to angle DEF.” Arc marks are used to show that the two angles are congruent. Example: Use arc marks to show that
∠PXR ∠QXS ≅
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Angle Addition Postulate
Bisector: a line that divides an _____________ ( ________________ ) into 2 congruent __________________ ( __________________ ).
Example 1: KM bisects ∠JKL, m∠JKM = (4x + 6), and m∠MKL = (7x – 12). Find m∠JKM.
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Example 2: If bisects ∠PXR and m∠PXQ = 25, then m∠ PXR = ________
1: E is between D and F. Find DF.
For #’s 4 and 5 refer to the diagram to the right:
4. bisects ∠QXS and m∠QXS = 60 ° then m∠RXS = ________
5. If bisects ∠PXR and m∠PXQ = 2Y + 15, and m∠QXR = 3y - 10, find y and m∠PXR.
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Lesson 1.3: Classify pairs of angles Warmup: Given AC in the diagram, state whether you can conclude the following (answer yes or no):
1. Points A, B, and C are collinear _______
2. Point B is between A and C _______ 3. AB = BC _______ 4. Point D is in the interior of ∠ABE _______ 5. m∠DBE = m∠EBC _______
6. bisects ∠DBC _______ Pairs of Angles: Many pairs of angles have special relationships. Some relationships are because of the measurements of the angles in the pair. Other relationships are because of the positions of the angles in the pair. Adjacent Angles are two angles in the same plane with a common vertex and a common side, but no common interior points. ∠1 and ∠2 are adjacent.
➢ Adjacent means next to A Linear Pair of angles is a pair of adjacent angles whose noncommon sides are opposite rays. ∠3 and ∠4 are a linear pair. (the angles form a straight line)
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Examples: Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent.
1. ∠AEB and ∠BED: 2. ∠AEB and ∠BEC: 3. ∠DEC and ∠AEB: If 2 angles are ______________________ then the sum of their degree measures
is ______________. 2 1 If 2 angles are ______________________ then the sum of their degree measures
is ______________. If 2 lines are _________________, then they meet to form a ________ _______.
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Vertical angles are two nonadjacent and congruent angles formed by two intersecting lines. ∠1 and ∠3 are vertical angles, as are ∠2 and ∠4. Name the pairs of vertical angles.
If 2 angles are _______________ _________ then they are _______________. Mark the pairs of congruent angles on the diagram below: 2 1 3 4 Examples: For examples 1-3, solve for x.
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Find the measure of each of the following. 4. complement of ∠F 5. supplement of ∠G 6. complement of ∠E 7. supplement of ∠F
Word Problems: 8. An angle’s measure is 6 degrees more than 3 times the measure of its complement.
Find the measure of the angle.
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9. ∠ABD and ∠DBE are supplementary. m∠ABD = 5x and m∠DBE = 17x – 18. Find the measure of both angles.
10. ∠ABD and ∠DBC are complementary. m∠ABD = 5y + 1 and m∠DBC = 3Y - 7.
Find the measure of both angles. Think about it… 11a) Are any two right angles supplementary? 11b) Can a pair of vertical angles also be adjacent?
11c) Explain why any two right angles are congruent.
11d) BD bisects ∠ABC. How are m∠ABC, m∠ABD and m∠DBC related?
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Lesson 1.4: Angles formed by lines intersected by a transversal Definitions: Parallel lines:____________________________________________________ ______________________________________________________________ Perpendicular lines:_______________________________________________ ______________________________________________________________ Transversal:_____________________________________________________ ______________________________________________________________ Example 1: Using the diagram below, identify each of the following:
1) a pair of parallel segments
2) a transversal
3) a pair of perpendicular segments
Angles formed by lines and a transversal examples: Corresponding Angles: <2 and <6 Alternate Interior Angles: <4 and <5 Alternate Exterior Angles: <2 and <7 Same Side Interior Angles: <4 and <6
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Practice: #1.Give an example of each angle pair:
1) corresponding angles
2) alternate interior angles
3) alternate exterior angles
4) same-side interior angles #2. Name the appropriate angle relationship for each pair of angles:
1) 1 and 3∠ ∠
2) 3 and 6∠ ∠
3) 4 and 5∠ ∠
4) 6 and 7∠ ∠
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Corresponding angles postulate: If two _______________ lines are cut by a transversal, then the ______________ angles are congruent. According to this postulate, which pairs of angles are congruent?
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Alternate Interior Angles Theorem: If two _______________ lines are cut by a transversal, then the pairs of _________________ _____________ angles are ______________. According to this theorem, which pairs of angles are congruent?
Alternate Exterior Angles Theorem: If two _______________ lines are cut by a transversal, then the pairs of _________________ _____________ angles are ______________.
According to this theorem, which pairs of angles are congruent? Same-side Interior Angles Theorem: If two _______________ lines are cut by a transversal, then the pairs of ________ _________ _____________ angles are _____________________. According to this theorem, which pairs of angles are supplementary?
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Examples: 1) Find the m<ABC:
2) Find the m<DEF.
3) a) Find the m<EDG. b) Find the m<BDG.
4) Find x and y:
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Lesson 1.5: Midpoint Review Draw a picture of a coordinate plane. Don’t forget to label your axes. Midpoint: You can find the midpoint of a segment by using the coordinates of its endpoints. Calculate the average of the x-coordinates and the average of the y - coordinates of the endpoints. How do you find the average of a set of numbers?
● Draw a line segment on the grid above with endpoints (-6,4) and (0,2).
● What is the average of the x-coordinates? ______
● What is the average of the y-coordinates? ______
● The Midpoint is (___,___)
General formula for finding the midpoint: Example: Find the midpoint of AB given A(-3, 7), B(2, 5).
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Lesson 1.6: Radicals Review
Index (small number outside the symbol)
Radicand (big number under the symbol)
Radical symbol without an index asks you to find the square root of a number. (The square root is the 2nd root)
Since (3)(3) = 9, Perfect square numbers are 1, 4, 9, 16, ___________________________ When you take the square root of a perfect square number, you will have a rational number. Find these square roots:
Most square roots are not perfect square roots, since most numbers are not perfect squares. When this happens, the number is irrational. Irrational numbers are decimals that do not ever end or repeat. Try these with your calculator.
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SIMPLIFYING RADICALS
When there is an irrational number we like to simplify it just as we like to reduce fractions to lowest terms.
A radical can be simplified when the number under the radical has a
perfect square as one of its factors. Try factoring these numbers, making sure one of the factors is a perfect square number that is not 1. 12 50 8 98 To simplify a radical, follow the steps
● Write as two radicals with the perfect square number first ● Write the square number in front of the radical.
(Check on your calculator that ) Try simplifying these:
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1) Find the midpoint of AB given A(7, 5) and B(-3, 9)
2) Can you exchange the coordinates (x1, y1) and (x2, y2) in the Midpoint formula and still find the correct midpoint? Explain.
3) Simplify
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Lesson 1.7: Distance Formula Warmup: Find these square roots:
Simplify the following radicals:
The Pythagorean Theorem In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.
a2 + b2 = c2
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In a coordinate plane, the distance d between two points (x1,y1) and (x2,y2) is NOTE: No negative distances! REMEMBER: When you square a number, it cannot be negative. Example 1: Find the coordinates of points F, G, J and K and the lengths of FG and JK. Example 2: Find the length of given E(-2,1) and F(-5,5).EF
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Example 3: Find the length of given A(0,3) and B(5,1).AB
Example 4: A player throws the ball from first base to a point located between third base and home plate and 10 feet from third base. What is the distance of the throw, to the nearest tenth?
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Lesson 1.8: Classify Triangles, Medians & Altitudes Triangles can be classified in different ways. In this exploration, you will sort triangles according to their angle measures and side lengths. Look at these triangles. Based on their appearance, sort the triangles by listing them in the appropriate columns. A triangle may be listed in more than one column.
AB, and are the sides of ΔABC.BC AC
A, B and C are the triangle’s vertices.
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Classifying Triangles: Recall that a triangle ( ) is a polygon with three sides. Triangles can be classified in two ways: by their angle measures or by their side lengths. Angle Measures Classify ΔFHG by its angle measures. Classify ΔFEH by its angle measures. Side Lengths
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Classify ΔABD by its side lengths. Classify ΔABC by its side lengths.
Remember, when you look at a figure, you cannot assume segments are congruent based on appearance. They must be marked as congruent.
Find the side lengths of ΔJKL. JK = _____ KL = _____ JL = _____ Find the side lengths of equilateral ΔFGH.
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The ___________ of a ________________ is a segment whose endpoints are a ___________ of the triangle and the _________ of the opposite side. Draw approximate medians in the following triangles. CD An ____________________ of a triangle is a perpendicular segment from a vertex to the opposite side. Sometimes altitudes are not inside the triangle. How come? Investigate below.
1) An acute triangle has ________________________________________ 2) An equiangular triangle has ____________________________________ 3) An isosceles triangle has ______________________________________
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Lesson 1.9: Angle Sum and Exterior Angle Theorems Triangle Angle-Sum Theorem If the measures of two angles of a triangle are known, the measure of the third angle can always be found. Triangle Angle Sum Theorem
The sum of the measures of the angles of a triangle is 180. In the figure at the right, m∠A + m∠B + m∠C = 180.
Example 1: Find m∠T. Example 2: Find the missing angle measures.
Exercises
Find the measure of each numbered angle. 1. 2.
3. 4. 5. 6.
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Exterior Angle Theorem: At each vertex of a triangle, the angle formed by one side and an extension of the other side is called an exterior angle of the triangle. For each exterior angle of a triangle, the remote interior angles are the interior angles that are not adjacent to that exterior angle. In the diagram below, ∠B and ∠A are the remote interior angles for exterior ∠DCB. Exterior Angle Theorem
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. m∠1 = m∠A + m∠B
Example 1: Find m∠1. Example 2: Find x.
Exercises
Find the measures of each numbered angle. 1. 2. 3. 4. Find each measure. 5. m∠ABC 6. m∠F
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Lesson 1.10: Triangle Inequality Theorems Triangle Inequality Theorem 1:
The sum of the lengths of any two sides of a triangle must be _________________________
__________________________________.
Examples: 1] Which of the following could represent the lengths of the sides of a triangle? Choose one: (1) 1, 2, 3 (2) 6, 8, 15 (3) 5, 7, 9 2] Two sides of an isosceles triangle measure 3 and 7. Which of the following could be the measure of the third side ? Choose one: (1) 9 (2) 7 (3) 3 Triangle Inequality Theorem 2:
In a triangle, the longest side is always across from the largest _________________.
Example:
Since 7 is the longest side in the triangle, <C, across from it, is the largest
angle.
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Triangle Inequality Theorem 3: (Opposite of Theorem 2 is true)
In a triangle, the largest angle is always across from the longest ______________________.
Example:
Since 100° is the largest angle in this triangle, the side across from it, is the longest side.AB
Triangle Inequality Theorem 4:
The measure of the exterior angle of a triangle is greater than the measure of either
nonadjacent interior angle.
Example: _______________________
_______________________
<1 is the exterior angle. <2 and <3 are its nonadjacent interior angles. Practice: (DRAW PICTURES TO HELP YOU) 1] In m<A = 30 and m<B = 50. Which is the longest side of the triangle?ABC, Δ Choose one: a) AB b) BC
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c) AC 2] In DEF, an exterior angle at D measures 170°, and m<E = 80. Which is theΔ longest side of the triangle ? Choose one: a) EF b) DE c) DF 3] In ABC, m<C = 55, and m<C > m<B. Which is the longest side of the triangle?Δ Choose one: a) AB b) AC c) CB
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