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Ch. 4 – Triangle Congruence (Notes-Renes-Honors Geometry)
4-2 Classifying Triangles
Warm Up
Draw and Classify each angle as acute, obtuse, or right.
1.
2.
3.
4. If the perimeter is 47, find x and the length
Objectives
 Classify triangles by their angle measures and side lengths.
 Use triangle classification to find angle measures and side lengths.
Vocabulary
acute triangle
equiangular triangle
right triangle
obtuse triangle
equilateral triangle
isosceles triangle
scalene triangle
***Recall that a triangle is a polygon with ___ sides. Triangles can be classified in two ways: by their
_____________________ measures or by their __________________ lengths.
___ ___ ___ are the sides of ABC.
A, B, C are the triangle's __________.
Triangle Classification by ANGLE MEASURES
Ex 1) Classify  BDC by its angle measures.
Classify ABD by its angle measures.
Classify  FHG by its angle measures.
Triangle Classification by SIDE LENGTHS
***Remember***When you look at a figure, you cannot assume segments are congruent based on appearance. They
must be marked as congruent.
Ex 2) Classify EHF by its side lengths
Classify  EHG by its side lengths.
Classify  ACD by its side lengths.
Example 3: Using Triangle Classification
Find the side lengths of JKL.
Find the side lengths of equilateral  FGH.
Example 4: 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?
Each measure is the side length of an equilateral triangle. Determine how many 10 in. triangles can be formed from a
100 in. piece of steel.
4-3 Angle Relationships in Triangles
Warmup
1. Find the measure of exterior DBA of  BCD, if mDBC = 30°, mC= 70°, and mD = 80°.
2. What is the complement of an angle with measure 17°?
3. How many lines can be drawn through N parallel to MP? Why?
Objectives
 Find the measures of interior and exterior angles of triangles.
 Apply theorems about the interior and exterior angles of triangles.
Vocabulary
auxiliary line
corollary
interior
exterior
interior angle
exterior angle
remote interior angle
An auxiliary line is a line that is added to a figure to aid in a proof.
Example 1A: 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 mXYZ.
Use the diagram to find mMJK.
A corollary is a theorem whose proof follows directly from another theorem. Here are two corollaries to the
Triangle Sum Theorem.
Example 2: Finding Angle Measures in Right Triangles
One of the acute angles in a right triangle measures 2x°. What is the measure of the other acute angle?
The measure of one of the acute angles in a right triangle is 63.7°. What is the measure of the other acute angle?
The measure of one of the acute angles in a right triangle is x°. What is the measure of the other acute angle?
2
The measure of one of the acute angles in a right triangle is 485. What is the measure of the other acute angle?
The interior is the set of all points inside the figure. The exterior is the set of all points outside the figure.
An interior angle is formed by two sides of a triangle.
An exterior angle is formed by one side of the triangle and extension of an adjacent side.
Each exterior angle has two remote interior angles. A remote interior angle is an interior angle that is not
adjacent to the exterior angle.
Example 3: Applying the Exterior Angle Theorem
Find mB
Find mACD
Example 4: Applying the Third Angles Theorem
Find mK and mJ
Find mP and mT
Warm Up
1. Name all sides and angles of ∆FGH.
2. What is true about K and L? Why?
3. What does it mean for two segments to be congruent?
4-4 Congruent Triangles
Objectives
 Use properties of congruent triangles.
 Prove triangles congruent by using the definition of congruence.
Vocabulary
 corresponding angles
 corresponding sides
 congruent polygons
Geometric figures are congruent if they are the same size and shape. Corresponding angles and corresponding
sides are in the same position in polygons with an equal number of sides. Two polygons are congruent
polygons if and only if their corresponding sides are congruent. Thus triangles that are the same size and shape
are congruent.
Two vertices that are the endpoints of a side are called _____________________
For example ,_____ and _____ are consecutive vertices.
___________________
To name a polygon, write the vertices in consecutive order. For example, you can name polygon PQRS as QRSP
or SRQP, but not as PRQS.
In a congruence statement, the order of the vertices indicates the corresponding parts.
***When you write a statement such as ABC  DEF, you are also stating which parts are congruent.***
Example 1: Naming Congruent Corresponding Parts
-Given: ∆PQR  ∆STW
Identify all pairs of corresponding congruent parts.
Angles:
Sides:
-If polygon LMNP  polygon EFGH, identify all pairs of corresponding congruent parts.
Example 2A: Using Corresponding Parts of Congruent Triangles
Given: ∆ABC  ∆DBC. Find the value of x.
Find mDBC.
Given: ∆ABC  ∆DEF
Find the value of x.
Find mF
Example 3: Proving Triangles Congruent
Given: YWX and YWZ are right angles.
YW bisects XYZ. W is the midpoint of XZ. XY  YZ.
Prove: ∆XYW  ∆ZYW
Given: AD bisects BE.
BE bisects AD.
AB  DE, A  D
Prove: ∆ABC  ∆DEC
Proof
Statements
Reasons______
Proof
Statements
Reasons______
Example 4: Engineering Application
The diagonal bars across a gate give it support. Since the angle measures and the lengths of the corresponding
sides are the same, the triangles are congruent.
Given: PR and QT bisect each other.
PQS  RTS, QP  RT
Prove: ∆QPS  ∆TRS
Proof
Statements
Reasons____________
Given: MK bisects JL. JL bisects MK. JK  ML. JK || ML.
Prove: ∆JKN  ∆LMN
Statements
Proof
Reasons____________
4-5 Triangle Congrunece: SSS and SAS
Warm Up
1. Name the angle formed by ray AB and ray AC.
2. Name the three sides of ABC.
3. ∆QRS  ∆LMN. Name all pairs of congruent corresponding parts.
Objectives
 Apply SSS and SAS to construct triangles and solve problems.
 Prove triangles congruent by using SSS and SAS.
Vocabulary
- triangle rigidity
- included angle
In Lesson 4-3, you proved triangles congruent by showing that all six pairs of corresponding parts were
congruent.
The property of triangle rigidity gives you a shortcut for proving two triangles congruent. It states that if the
side lengths of a triangle are given, the triangle can have only one shape.
Adjacent triangles share a side, so you can apply the Reflexive Property to get a pair of congruent parts.
Draw:
Example 1: Using SSS to Prove Triangle Congruence
Use SSS to explain why ∆ABC  ∆DBC.
Use SSS to explain why ∆ABC  ∆CDA.
An included angle is an angle formed by two adjacent sides of a polygon.
B is the included angle between sides AB and BC.
The letters SAS are written in that order because the congruent angles must be between pairs of congruent
corresponding sides!!!
Example 2: Engineering Application
The diagram shows part of the support structure for a tower. Use SAS to explain why ∆XYZ  ∆VWZ.
Use SAS to explain why ∆ABC  ∆DBC.
The SAS Postulate guarantees that if you are given the lengths of two sides and the measure of the included angles, you
can construct one and only one triangle.
Example 3: Verifying Triangle Congruence
Show that the triangles are congruent for the given value of the variable.
∆MNO  ∆PQR, when x = 5.
∆STU  ∆VWX, when y = 4.
Show that ∆ADB  ∆CDB, t = 4.
Given: BC ║ AD, BC  AD
Prove: ∆ABD  ∆CDB
Given: ray QP bisects RQS. QR  QS
Prove: ∆RQP  ∆SQP
Proof
Statements
Reasons____________
Proof
Statements
Reasons__________________
Warm Up
1. What are sides AC and BC called? Side AB?
2. Which side is in between A and C?
3. Given DEF and GHI, if D  G and E  H, why is F  I?
4-6 Triangle Congrunece: ASA , AAS, and HL
Objectives
- Apply ASA, AAS, and HL to construct triangles and to solve problems.
- Prove triangles congruent by using ASA, AAS, and HL.
Vocabulary
- Included Side
An included side is the common side of two consecutive angles in a polygon. The following postulate uses the
idea of an included side.
Participants in an orienteering race use a map and a compass to find their way to checkpoints along an unfamiliar
course.
Directions are given by bearings, which are based on compass headings. For example, to travel along the bearing S 43°
E, you face south and then turn 43° to the east.
Example 1: Problem Solving Application
A mailman has to collect mail from mailboxes at A and B and drop it off at the post office at C. Does the table
give enough information to determine the location of the mailboxes and the post office?
Example 2: Applying ASA Congruence
Determine if you can use ASA to prove the triangles congruent. Explain.
Determine if you can use ASA to prove NKL  LMN. Explain.
You can use the Third Angles Theorem to prove another congruence relationship based on ASA. This theorem
is Angle-Angle-Side (AAS).
Example 3: Using AAS to Prove Triangles Congruent
Use AAS to prove the triangles congruent.
Given: X  V, YZW  YWZ, XY  VY
Prove:  XYZ  VYW
Given: JL bisects KLM, K  M
Prove: JKL  JML
Example 4A: Applying HL Congruence
Determine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell what else
you need to know.
Determine if you can use the HL Congruence Theorem to prove ABC  DCB. If not, tell what else you need
to know.
4-7 Triangle Congruence: CPCTC
Warm Up
1. If ∆ABC  ∆DEF, then A  ? and BC  ? .
3. If 1  2, why is a||b?
2. What is the distance between (3, 4) and (–1, 5)?
4. List methods used to prove two triangles
congruent.
Objectives
 Use CPCTC to prove parts of triangles are congruent.
***CPCTC is an abbreviation for the phrase “C______________ P_________ of C_______________
T_____________ are C________________.” It can be used as a justification in a proof
after you have
proven two triangles congruent.
Example 1: Engineering Application
A and B are on the edges of a ravine. What is AB?
A landscape architect sets up the triangles shown in the figure to find the distance JK
across a pond. What is JK?
Example 2: Proving Corresponding Parts Congruent
Given: YW bisects XZ, XY  YZ.
Prove: XYW  ZYW
Example 3: Using CPCTC in a Proof
Given: PR bisects QPS and QRS.
Prove: PQ  PS
***Work backward when planning a proof. To show that ED || GF, look for a pair of angles that are congruent.
Then look for triangles that contain these angles.***
Given: NO || MP, N  P
Prove: MN || OP
___Statements
Given: J is the midpoint of KM and NL.
___Statements
Reasons____
Prove: KL || MN
Reasons____
Example 4: Using CPCTC In the Coordinate Plane
Given: D(–5, –5), E(–3, –1), F(–2, –3), G(–2, 1), H(0, 5), and I(1, 3)
Prove: DEF  GHI
4-7-ext Lines and Slopes
Objective :
Prove the slope criteria for parallel and perpendicular lines.
Slopes can be used to determine if two lines in a coordinate plane are parallel or perpendicular. In this lesson, you will prove the
Parallel Lines Theorem and the Perpendicular Lines Theorem. Suppose that L1 and L2 are two lines in the coordinate plane with
slopes m1 and m2. The proof of the Parallel Lines Theorem can be broken into three parts:
1.
2.
3.
Example 1: Proving the Parallel Lines Theorem
Are the lines parallel? Explain.
Complete the two-column proof, using the figure in Example 1
Statements
Given: m1 = m2
Prove: L1 || L2
Reasons
Example 2 : Proving the Perpendicular Lines
Statements
Are the lines perpendicular? Explain.
Reasons
4-8 Intro to Coordinate Proof
Objectives
-Position figures in the coordinate plane for use in coordinate proofs.
-Prove geometric concepts by using coordinate proof.
A coordinate proof is a style of proof that uses coordinate geometry and algebra. The first step of a coordinate proof is
to position the given figure in the plane. You can use any position, but some strategies can make the steps of the proof
simpler.
Once the figure is placed in the coordinate plane, you can use slope, the coordinates of the vertices, the Distance
Formula, or the Midpoint Formula to prove statements about the figure.
Write a coordinate proof.
Given: Rectangle ABCD with A(0, 0), B(4, 0), C(4, 10), and D(0, 10)
Prove: The diagonals bisect each other.
Use the information in Example 2 (p. 268) to write a coordinate proof showing that the area of ∆ADB is one half the
area of ∆ABC.
Proof: ∆ABC is a right triangle with height AB and base BC.
Position each figure in the coordinate plane and give the coordinates of each vertex.
-rectangle with width m and length twice the width
-right triangle with legs of lengths s and t
-a square with side length 4p in the coordinate plane and give the coordinates of each vertex.
4-9 Isosceles and Equilateral Triangles
Objectives


Prove theorems about isosceles and equilateral triangles.
Apply properties of isosceles and equilateral triangles.
Vocabulary
legs of an isosceles triangle
vertex angle
base
base angles
Recall that an isosceles triangle has at least _____ congruent sides. The congruent sides are called the ____. The
_____________ ____ is the angle formed by the legs. The side opposite the vertex angle is called the ________, and
the ________________ are the two angles that have the base as a side.
____is the vertex angle.
____and _____ are the base angles.
Example 1: Astronomy Application
The length of YX is 20 feet.
Explain why the length of YZ is the same.
Example 2A: Finding the Measure of an Angle
Find mF.
Find mG.
Find mN.
Find the value of x.
Find the value of y.
Example 4: Using Coordinate Proof
Prove that the segment joining the midpoints of two sides of an isosceles triangle is half the base.
Given: In isosceles ∆ABC, X is the mdpt. of AB, and Y is the mdpt. of AC.
Prove: XY = 1/2AC.