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CONGRUENT TRIANGLES Chapter 4 β Unit 6
SPRING 2019 GEOMETRY
Name ____________________________________Hour ______
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Geometry Classifying Triangles 4.1
Triangles can be classified by their _______________ and/or their ________________.
Triangle ABC use the symbol ABC
Each point is a _______________________.
Example 4-1-1: Classify each triangle by its angles: acute, equiangular, obtuse, or right.
Example 4-1-2: Classify the Triangle by angle measures
a. BDC
b. ABD
Example 4-1-3: Classify the Triangle by side lengths
a. EHF
b. EHG
Adjacent Sides C
Side opposite A
B A
leg
leg
hypotenuse
Objectives:
1) classify triangles by their angle measures and side lengths
2) use triangle classification to find angle measures and side lengths
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Geometry 1
Example 4-1-4: If point M is the midpoint of π½οΏ½Μ οΏ½, classify βππΏπΎ, βπ½πΎπ, and βπΎπ½πΏ by their sides: equilateral, isosceles, or scalene. Example 4-1-6: Find the measures of the sides of isosceles triangle ABC π΄π΅ = π΄πΆ = πΆπ΅ = Example 4-1-7: If the perimeter is 47, find x and the length of each side. π₯ =_____
π·πΈ =_____
π·πΉ =_____
πΈπΉ =_____
Example 4-1-8: Real World Application
A steel mill produces roof supports by welding pieces of steel beams into equilateral
triangles. Each side of the triangle is 18 feet long. How many triangles can be formed
from 420 feet of steel beam?
Think and Discuss
Why do you round the answer down instead of rounding the answer up?
D
E F
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Geometry 2
Geometry Angle Relationships in Triangles 4.2
WARM-UP 1. What is the complement of an angle with measure 17Β°? 2. How many lines can be drawn through a point not on a line, parallel to a given line? Why? Explore: Go to the last page and rip off the triangle, then rip two of the corners and place them similar to the example. What conclusion can you draw about the sum of the angles? ______________ What conclusion can you make about the exterior angle compared to two of the angles?
Triangle Sum Theorem β
Auxiliary line A line added to a diagram to help analyze the diagram. (Below:
π΄π· β‘ was added to make a line parallel to the π΅πΆΜ Μ Μ Μ by the Parallel Postulate.)
Whenever you draw an auxiliary line, you must be able to justify its existence. Give this as the reason: Through any two points there is exactly one line.
Objectives:
1) find the measures of interior and exterior angles of triangle
2) apply theorems about the interior and exterior angles of triangles
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Geometry 3
Example 4-2-1: Real World Application
After an accident, the positions of cars are measured by law enforcement to investigate the collision. Use the diagram drawn from the information collected to find the indicated angle measures.
a. m XYZ
b. π πππ
Example 4-2-2: Real World Application The diagram
shows the path of the softball in a drill developed by
four players. Find the measure of each numbered angle.
π 1 =
π 2 =
π 3 =
π 1 = π 4 = π 7 =
π 2 = π5 = π 8 =
π 3 = π 6 = π 9 =
Example 4-2-3: One of the acute angles in a right triangle measures 2xΒ°. What is the
measure of the other acute angle?
Exterior Angle the _________ formed by one side and an
_________ of the other side In the picture, it is angle ________
Remote Interior the interior angles that are not
____________ to that exterior angle In the picture, the angles are ____________
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Geometry 4
Example 4-2-4: Find the ππΉπΏπ in the fenced flower garden shown.
ππΉπΏπ = ___________ Corollary A theorem that can be proven directly from another theorem.
Corollary 4.1 β the acute angles in a right triangle are _________________
Corollary 4.2 β there can be at most one right or obtuse angle in a triangle.
Corollary β the angles in an Equiangular Triangle are ______________.
Complete the 2-column proof of Corollary 4.1: Given: Right triangle ABC Prove: β π΄ πππ β π΅ are complementary
Statements Reasons 1. 1.
2. πβ π΄ + πβ π΅ + πβ πΆ = 180 2.
3. πβ πΆ = 90 3. Def. of right triangle
4. πβ π΄ + πβ π΅ + 90 = 180 4.
5. 5.
6. 6.
Example 4-2-5: Find the mB.
π₯ = ___________
π π΅ = ___________
A
C B
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Geometry 5
Example 4-2-6: Find Angle Measures in Right Triangles
π 1 = π 2 = π 3 =
π 4 = π 5 =
Think and Discuss
Geometry Congruent Triangles 4.3
WARM-UP
State the property that justifies each statement.
1. π΄π΅ = π΄π΅.
2. If πΈπΉ = πΊπ» and πΊπ» = π½πΎ, then πΈπΉ = π½πΎ.
3. If π2 = π2 β π2, then π2 β π2 = π2.
4. If ππ + 20 = ππ and ππ + 20 = π·π, then ππ = π·π.
5. What does it mean for two segments to be congruent?
Two figures are ____________________ when they have corresponding angles and
corresponding sides that are congruent.
Objectives:
1. use properties of
congruent triangles
2. prove triangles
congruent by using
the definition of
congruence
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Geometry 6
Geometry Congruent Triangles 4.3
Example 4-3-1: Identify Corresponding Congruent Parts
Show that the polygons are congruent by identifying all of
the congruent corresponding parts. Then write a
congruence statement.
Sides Angles
Triangle Congruence (CPCTC)
Knowing that all pairs of corresponding parts of congruent triangles are congruent
(CPCTC) can help you reach conclusions about congruent figures.
CPCTC Corresponding Parts of Congruent Triangles are Congruent
Example 4-3-2: Use Corresponding Parts of Congruent Triangles
In the diagram, ΞITP ΞNGO. Find the values of
x and y.
Example 4-3-3: Given: ABC DBC
a. Find the value of x
b. Find the π π·π΅πΆ
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Geometry 7
Example 4-3-4: Proving triangles congruent.
Given: YWX and YWZ are right angles. YW bisects
XYZ. W is the midpoint of XZ . YZXY
Prove: ZYWXYW
Statements Reasons
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Example 4-3-5: Use the Third Angles Theorem
A drawing of a towerβs roof is composed of congruent triangles
all converging at a point at the top. If IJK IKJ and mIJK =
72, find mJIH.
Example 4-3-6: Find the mK and mJ.
First find π¦2 =
π πΎ = ____________ π π½
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Geometry 8
Example 4-3-6: Prove Triangles Congruent
Given: π½οΏ½Μ οΏ½ β ππΏΜ Μ Μ Μ , and L bisects πΎπΜ Μ Μ Μ Μ .
β π½ β β π, π½πΎΜ Μ Μ β ππΜ Μ Μ Μ Μ
Prove: Ξ π½πΏπΎ β Ξ ππΏπ
Statements Reasons
1 1. Given
2 2.
3 3.
4 4.
5 5.
6 6.
7 7.
Think and Discuss β complete the graphic organizer
Geometry Proving Triangles are Congruent: SSS and SAS 4.4
WARM-UP
1. Name the angle formed by π΄π΅ and π΄πΆ . 2. Name the three sides of βπ΄π΅πΆ. 3. βππ π β βπΏππ. Name all pairs of congruent corresponding parts.
4. What are sides π΄πΆ and π΅πΆ called? side π΄π΅? 5. Which side is between β π΄ and β πΆ?
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Geometry 9
Geometry Proving Triangles are Congruent: SSS and SAS 4.4
Triangle Rigidity β
Example 4-4-1: Use SSS to Prove Triangles Congruent
Given: ππΜ Μ Μ Μ β π΄π·Μ Μ Μ Μ , ππ·Μ Μ Μ Μ β π΄πΜ Μ Μ Μ
Prove: Ξ πππ· β Ξ π΄π·π
Statements Reasons
1. 1. Given
2. 2.
3. 3.
Included Angle Example 4-4-2: Use SAS to Prove Triangles are Congruent
The wings of one type of moth form two triangles. Write a
two-column proof to prove that π₯πΉπΈπΊ π₯π»πΌπΊ, if πΈπΌΜ Μ Μ πΉπ»Μ Μ Μ Μ ,
and G is the midpoint of both πΈπΌΜ Μ Μ and πΉπ»Μ Μ Μ Μ .
Statements Reasons
1. 1. Given
2. 2.
3. 3.
4. 4.
Objectives:
1. apply SSS and SAS
to construct
triangles and to
solve problems
2. prove triangles
congruent by using
SSS and SAS to
construct triangles
and to solve
problems.
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Geometry 10
Example 4-4-4: Show that the triangles are congruent for the given value of the variable.
a. MNO PQR, when x = 5
b. STU VWX, when y = 4
Example 4-4-5: Determine if SSS or SAS or neither is used in the following triangle congruency. Mark your diagram! Ex A: Ex B: Ex C:
Given: A is the midpoint of π½πΜ Μ Μ Μ Given: πΎπ΄Μ Μ Μ Μ bisects β π½πΎπ Given: β π½ β β π
Example 4-4-6:
Given: π΅πΆΜ Μ Μ Μ ||π΄π·Μ Μ Μ Μ , π΅πΆΜ Μ Μ Μ β π΄π·Μ Μ Μ Μ
Prove: Ξ π΄π΅π· β Ξ πΆπ·π΅
Statements Reasons
1. 1. Given
2. 2. Given
3. 3.
4. 4.
5. 5.
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Geometry 11
Example 4-4-7:
Given: ππΜ Μ Μ Μ Μ β ππΜ Μ Μ Μ , πΏπΜ Μ Μ Μ β πΏπΜ Μ Μ Μ
Prove: β πΏππ β β πΏππ
Statements Reasons
1. 1.
2. 2.
3. ΞLNM β βπΏππ 3.
4. 4. CPCTC
Example 4-4-8: Real World Application
A and B are on the edges of a ravine. What is AB?
Think and Discuss β complete the graphic organizer
Geometry Proving Triangles are Congruent: ASA and AAS 4.5
WARM-UP In the diagram, β πΏππ β β ππ π
1. Find x. 2. Find y.
Write an equation in slope-intercept form for each line.
3. (β5, β3) and (10, β6)
π¦ = _____π₯ +_____ 4. (4, β1) and (β2, β1)
π¦ = _____π₯ +_____
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Geometry 12
Geometry Proving Triangles are Congruent: ASA and AAS 4.5
Included side
NOTE: Two methods that CANNOT be used are: AAA and SSA
Example 4-5-1 β Use ASA to Prove Triangles Congruent
Given: ππ Μ Μ Μ Μ Μ ||πΈπ·Μ Μ Μ Μ , πΏ is the midpoint of ππΈΜ Μ Μ Μ Μ Prove: Ξ ππ πΏ β Ξ πΈπ·πΏ
Example 4-5-2 β Use AAS to Prove Triangles Congruent
Given: πΎπΏΜ Μ Μ Μ β π½πΜ Μ Μ Μ , β ππΎπΏ β β ππ½π Prove: πΏπΜ Μ Μ Μ β ππΜ Μ Μ Μ Μ
Objectives:
1. apply ASA, AAS,
and HL to construct
triangles and to
solve problems
2. prove triangles
congruent by using
ASA, AAS, and
HL to construct
triangles and to
solve problems.
Think about Example 4-5-1 We used ASA, can it be done with AAS? If not, why not. If so, what additional information would you need?
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Geometry 13
Example 4-5-3: Use AAS to prove the triangles congruent
Given: VX , YWZYZW , VYXY
Prove: VYWXYZ
Example 4-5-4: Which method would you use? Support your reasoning with correct markings on the diagram. A. B.
C. Given: π½πΏ bisects β πΎπΏπ. β πΎ β β π.
Example 4-5-5: Flow Proof
Given: π πΜ Μ Μ Μ ||ππΜ Μ Μ Μ , π πΜ Μ Μ Μ β ππΜ Μ Μ Μ
Prove: Ξ π ππ β Ξ πππ
M
LJ
K
Plan: 1st possible approach: Use a parallel line theorem and vertical angles to use AAS. 2nd possible approach: Use parallel line theorem twice to use ASA.
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Geometry 14
4.5 β Extension: Right Triangle Congruence: LL, HA, LA, HL
Example 4-5-6: Determine if you can use the HL Congruence Theorem to prove the
triangles congruent. If not, tell what else you need to know.
Example 4-5-7:
Given: π΄π΅Μ Μ Μ Μ β₯ π΅πΆΜ Μ Μ Μ , π·πΆΜ Μ Μ Μ β₯ π΅πΆΜ Μ Μ Μ , π΄πΆΜ Μ Μ Μ β π΅π·Μ Μ Μ Μ
Prove: ABΜ Μ Μ Μ β π·πΆΜ Μ Μ Μ
Statements Reasons
1. 1.
2. 2.
3. 3.
4. 4.
5. 5.
6. 6.
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Geometry 15
Example 4-5-8: Determine whether each pair of triangles is congruent. If yes, tell which
postulate or theorem applies.
A. B. C.
D. E.
Think and Discuss β Complete the graphic organizer
Def of SSS SAS ASA AAS HL
Words
Pictures
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Geometry 16
Geometry Isosceles and Equilateral Triangles 4.6
WARM-UP: 1. Find each angle measure in the triangle. True or False. If false, explain. 2. Every equilateral triangle is isosceles. 3. Every isosceles triangle is equilateral.
Isosceles triangle An isosceles triangle has two congruent sides called the legs. The
angle formed by the legs is called the vertex angle. The other two angles are called base
angles. You can prove a theorem and its converse about isosceles triangles.
In this diagram,
the base angles are _____ and _____; the vertex angle is _______.
Example 4-6-1: Congruent Segments and Angles
A. Name two unmarked congruent angles.
B. Name two unmarked congruent segments.
Objectives:
1. prove theorems
about isosceles and
equilateral triangles
2. apply properties of
isosceles and
equilateral triangles
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Geometry 17
Example 4-6-2: Find Missing Measures
A. Find mR.
B. Find PR.
Example 4-6-3: Find Missing Values
Find the value of each variable.
Example 4-6-4: The length of YX is 20 feet. Explain why the
length of YZ is the same.
Example 4-6-5: Find each angle measure.
a. m F b. mG
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Geometry 18
Example 4-6-6: β Find each value.
a. x b. y
Example 4-6-8:
Given: YW bisects XZ YZXY
Prove: ZYWXYW
Example 4-6-9: Using CPCTC with another Thm.
Given: PNMPNO ,||
Prove: OPMN ||
X
Y
W Z
M
N O
P
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Geometry 19
Geometry Coordinate Proof 4.8
WARM-UP: Do the proof on the back page.
Example 4-8-1: Proof in the Coordinate Plane
Given: π·(β5, β5), πΈ (β3, β1), πΉ (β2, β3), πΊ(β2, 1), π» (0, 5) and πΌ (1, 3)
Prove: β π·πΈπΉ β β πΊπ»πΌ
π·πΈ = β(β1 β β5)2 + (β3 β β5)2
πΈπΉ =
πΉπ· =
πΊπ» =
π»πΌ =
πΌπΊ =
Strategies for doing Coordinate Proof
1. Use origin as a vertex
2. center figure at the origin
3. center side of figure at origin
4. use axes as sides of figure
Example 4-8-2: Position and Label a Triangle
Position and label right triangle XYZ with leg
d units long on the coordinate plane.
Example 4-8-3: Identify Missing Coordinates
Name the missing coordinates of isosceles right triangle
QRS.
Objectives:
1. Position and label
triangles for use in
coordinate proofs.
2. Write coordinate
proofs.
Apply the distance formula to all 3 sides of each triangle:
β(y2 β π¦1)2 + (π₯2 β π₯1)2
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Geometry 20
Example 4-8-4: SSS on the Coordinate Plane
Triangle DVW has vertices π·(β 5, β 1), π(β 1, β 2), and π(β 7, β 4). Triangle LPM has
vertices πΏ(1, β 5), π(2, β 1), and π(4, β 7).
a) Graph both triangles on the
same coordinate plane.
b) Use your graph to make a
conjecture as to whether the
triangles are congruent. Explain
your reasoning. Apply the distance formula to all 3 sides of each triangle:
β(y2 β π¦1)2 + (π₯2 β π₯1)2
π·π = β(β1 β β5)2 + (β2 β β1)2
ππ =
ππ· =
πΏπ =
ππ =
ππΏ =
c) Write a logical argument that uses
coordinate geometry to support the
conjecture you made in part b.
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Geometry 21
Example 4-8-5: Coordinate Proof
Prove that the segment joining the Midpoint of two sides of an isosceles triangle is half the base.
Given: In isosceles ABC, X is the midpoint of ABΜ Μ Μ Μ , and Y is the midpoint of ACΜ Μ Μ Μ .
Prove: XY =1
2 AC
Example 4-5-6: Write a Coordinate Proof
Write a coordinate proof to prove that the segment
that joins the vertex angle of an isosceles triangle
to the midpoint of its base is perpendicular to the
base.
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Extra Proof
Given: π΄π΅Μ Μ Μ Μ || π·πΆΜ Μ Μ Μ , π΄π΅Μ Μ Μ Μ β₯ π΅πΆΜ Μ Μ Μ , E is the midpoint of π΄πΆΜ Μ Μ Μ and π΅π·Μ Μ Μ Μ
Prove: ACΜ Μ Μ Μ β π·π΅Μ Μ Μ Μ
Statements Reasons
1. 1.
2. 2.
3. 3.
4. 4.
5. 5.
6. 6.
7. 7.
8. 8.
A
B
C