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Ch 5: Relationships Within Triangles
5‐1 Midsegments of Triangles
5‐2 Bisectors in Triangles
5‐3 Concurrent Lines, Medians, and Altitudes
5‐4 Inverses, Contrapositives, and Indirect Reasoning
5‐5 Inequalities in Triangles
5‐1 Midsegments of Triangles:
Focused Learning Target: I will be able to
Use properties of midsegments find lengths
Use properties of midsegments to identify parallel segments.
Standards: Geometry 17.0. Students prove theorems by using coordinate geometry
Vocabulary:
Midsegment
Coordinate proof
Midsegment: a midsegment of a triangle is a segment that connects the midpoints of two sides.
Coordinate Proof: a coordinate proof uses algebras and coordinate geometry to prove statements.
I’ll prove the Triangle Midsegment Theorem using a coordinate proof:
1. Begin by placing the triangle in a convenient location on the coordinate plane. 2. Use coordinate geometry and algebra to complete the proof.
Given: R is the midpoint of .QP
S is the midpoint of .QP
Prove: RS OQ and 1
2RS OQ
1. Find the midpoints R & S.
2. Show (using the distance formula) that 1
2RS OQ
Use the slope formula to show that RS OQ . (If the slopes are equal, then they are parallel.)
x
y
O(0,0) Q(a,0)
P(b,c)
R S
2
Using the Triangle Midsegment Theorem to find lengths: I’ll do one:
In ,EFG H, J, and K are midpoints. Find: HJ, JK, and FG.
We’ll do one together:
Find the value of each variable.
You Try:
Find the value of x.
Using the Triangle Midsegment Theorem to identify parallel segments: I’ll do one:
In ,DEF A, B and C are midpoints. Name pairs of parallel segments.
We’ll do one:
Name the pairs of parallel segments in FHK
3
You Try:
Name the segment that is parallel to each given segment.
AF
FE
CB
5‐2 Bisectors in Triangles:
Focused Learning Target: I will be able to
Use properties of perpendicular bisectors and angle bisectors
Vocabulary:
Distance from a point to a line
Standards: Geo 2.0 Students write geometric proofs (master) Geo 4.0 Students prove basic theorems involving congruence (master) Geo 5.0 Students prove that triangles are congruent and they are able to use the concept of corresponding parts of congruent triangles (master)
Example: Using the Perpendicular Bisector Theorem: Find AC and DB
4
I’ll do one:
Given: CD
is the bisector of AB Prove: ADC BDC
1. 1.
2. 2.
3. 3.
4. 4.
5. 5.
6. 6.
We’ll do one together:
Given: TW
is the bisector of XZ Prove: XZW is an isosceles triangle
1. 1.
2. 2.
3. 3.
4. 4.
You Try:
Given: QS is the bisector of PR
Prove: PQR is an isosceles triangle
1. 1.
2. 2.
3. 3.
4. 4.
5
Using the Angle Bisector Theorem: I’ll do one:
What is the length of FD
We’ll do one together:
According to the diagram, how far is L from ? ?HK from HF
You Try:
What is the length of , &x JK JM
5‐3 Concurrent Lines, Medians, and Altitudes
Focused Learning Target: I will be able to
Identify properties of perpendicular bisectors and angle bisectors.
Identify properties of medians and altitutdes of a triangle.
6
Vocabulary:
Concurrent
Point of concurrency
Circumcenter of a triangle
Circumscribed about
Incenter of a triangle
Inscribed in
Median of a triangle
Centroid
Altitude of a triangle
Altitude of a triangle
Standards: Geo 2.0: Students write geometric proofs (Master) Geo 21.0: Students prove and solve problems regarding relationships among inscribed and circumscribed polygons of circles (Introduce)
When three or more lines intersect at one point, they are concurrent. The point at which they intersect is the point of concurrency. For any triangle, four different sets of lines are concurrent. Theorems 5‐6 and 5‐7 tell you about two of them.
Identifying perpendicular bisectors: I’ll do one:
I’ll do one: You Try:
Identify the perpendicular bisector:
Identify the perpendicular bisector:
R
A
B
C
S
T
This figure shows QRS with the perpendicular bisectors of its sides concurrent at C. The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter of the triangle. Points Q, R, and S are equidistant from C, the circumcenter. The circle is circumscribed about the triangle. Any three non‐collinear points can be circumscribed.
7
Finding the circumcenter Coordinate Geometry I’ll do one: Find the center of the circle that you can circumscribe a circle about
.OPS
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
x
y
(2,-3)
(-2,3)
(-2,-3)
We’ll do one together: Find the center of the circle that you can circumscribe a circle about
.ABC
(0,0)
(4,0)
(4, 3)
A
B
C
-4 -3 -2 -1 1 2 3 4 5
-4
-3
-2
-1
1
2
3
x
y
You Try: Find the center of the circle that you can circumscribe a circle about
.ABC
( 2,2)
(3,2)
(3, 3)
A
B
C
-5 -4 -3 -2 -1 1 2 3 4 5 6
-4
-3
-2
-1
1
2
3
4
x
y
A(-2,2) B(3,2)
C(3,-3)
A median of a triangle is a segment whose endpoints are a vertex and the midpoint of the opposite side.
8
In a triangle, the point of concurrency of the medians is the centroid. The point is also called the center of gravity of a triangle because it is the point where a triangular shape will balance.
Finding lengths of Medians I’ll do one:
In ABC , D is the centroid and DE = 6, Find BE.
We’ll do one together:
In TUV , Y is the centroid. If YU = 9, Find ZY and ZU.
You Try:
In TUV , Y is the centroid. If XU = 39, Find XY and YV.
9
An altitude of a triangle is the perpendicular segment from a vertex to the line containing the opposite side. Unlike angle bisectors and medians, an altitude of a triangle can be a side of a triangle or it may lie outside the triangle.
Identifying Medians and Altitudes I’ll do one:
Is UW a median, or an altitude of VSU ?
You Try:
Name the altitude and the median in VSU . Perpenendicular bisector:
5‐4 Inverses, Contrapositives, and Indirect Reasoning
Focused Learning Target: I will be able to
Write the negation of a statement and the inverse and contrapositive of conditional statement.
Use indirect reasoning.
Standards: Geometry 2.0. Students write geometric proofs, including proofs by contradiction.
Vocabulary:
Negation: the negation of a statement has the opposite truth value
Inverse: the inverse of a conditional statement negates both hypothesis and conclusion.
Contrapositive: the contrapostive of conditional switches the hypothesis and conclusion and negates both.
Equivalent statements: have the same truth value.
Indirect reasoning: all possibilities are considered and all but one are proved false.
Indirect proof: proof involving indirect reasoning.
10
I’ll do one:
Write the negation of “ABCD is not a convex polygon” Answer:
We’ll do one together:
Write the negation of “Line m and n are not perpendicular” Answer:
You Try:
Write the negation of “ The restaurant is not open on Sunday” Answer:
I’ll do one:
Write the inverse and contrapositive of the conditional statement: Conditional: If two triangles are congruent, then their corresponding angles are congruent. Inverse: Contrapositive:
We’ll do one:
Write the inverse and contrapositive of the conditional statement:
Conditional: If ABC is equilateral, then it is isosceles. Inverse: Contrapositive:
11
You Try:
Write the inverse and contrapostive of the conditional statement:
Conditional: Inverse: Contrapositive:
I’ll do one:
Write the first step of an indirect proof: Prove: A triangle cannot contain two right angles. Answer:
We’ll do one together:
Write the first step of an indirect proof: Prove: Answer:
You Try:
Write the first step of an indirect proof: Prove: The shoes cost no more than $20. Answer:
I’ll do one:
Identify the two statements that contradict each other:
Answer:
We’ll do one together:
Identify the two statements that contradict each other:
Answer:
You Try:
Identify the two statements that contradict each other:
Answer:
12
I’ll do one:
Write an indirect proof: Prove: A triangle cannot contain two right angles. Answer:
We’ll do one together:
Write an indirect proof: Prove: An equilateral triangle cannot have a right angle. Answer:
You Try:
Write an indirect proof:
Prove: ABC cannot contain two obtuse angles. Answer:
5‐5 Inequalities in Triangles:
Focused Learning Target: I will be able to
Use inequalities involving angles of triangles
Use inequalities involving sides of triangles
Standards: Geo 2.0 Students write geometric proofs. Geo 6.0 Students know and be able to use the triangle inequality theorem.
Key Concepts:
13
I’ll do one:
We’ll do one together:
Explain why 1 2m m
You Try:
2 Example Using theorem 5‐10 I’ll do one:
Determine the two largest angle in each triangle:
14
Answer:
We’ll do one together:
Determine the two largest angle in each triangle:
Answer:
You Try:
Determine the two largest angle in each triangle:
Answer:
I’ll do one:
List the sides of each triangle in order from shortest to longest.
Answer:
We’ll do one together:
List the sides of each triangle in order from shortest to longest.
Answer:
You Try:
List the sides of each triangle in order from shortest to longest.
15
Answer:
I’ll do one:
Can a triangle have sides with the given length? Explain. a) 8 in., 12 in., and 15 in. b) 2 cm, 3 cm, and 6 cm Answer: a) b)
We’ll do one together:
Can a triangle have sides with the given length? Explain.
Answer:
You Try:
Can a triangle have sides with the given length? Explain.
Answer:
I’ll do one:
The lengths of two sides of a triangle are given. Describe the lengths possible for the third side. 8 ft., 12 ft. Answer:
We’ll do one together:
The lengths of two sides of a triangle are given. Describe the lengths possible for the third side. 9 cm, 17cm Answer:
You Try:
The lengths of two sides of a triangle are given. Describe the lengths possible for the third side. 5 ft., 5 ft. Answer: