Congruent Triangles Part 2 Unit 4 SOL G.5 & G.6 Sections 4.5, 4.6, 4.7 Review & Proofs Resource: Henrico Geometry

Parts of An Isosceles Triangle An isosceles triangle is a triangle with at least two congruent sides. The congruent sides are called legs and the third side is called the base. The angles adjacent to the base are called base angles. The angle opposite the base is called the vertex angle. and are legs. is the base. 1 and 2 are base angles. 3 is the vertex angle.

Isosceles Triangle Theorems If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

Example By the Isosceles Triangle Theorem, the third angle must also be x. Therefore, x + x + 50 = 180 2x + 50 = 180 2x = 130 x = 65 x

More Examples Since two angles are congruent, the sides opposite these angles must be congruent. 3x – 7 = x + 15 2x = 22 x = 11

Equilateral and Equiangular Triangles Corollary: If a triangle is equilateral, then … it is equiangular. .

What if all the ANGLES in a triangle are DIFFERENT? Then the triangle is SCALENE because all of the sides are DIFFERENT too!

The triangles are congruent by HL. Hypotenuse-Leg (HL) If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. Right Triangle Leg Hypotenuse AB  HL CB  GL C and G are rt.  ‘s ABC   DEF The triangles are congruent by HL.

Reflexive Sides and Angles Vertical Angles, Reflexive Sides and Angles When two triangles touch, there may be additional congruent parts. Vertical Angles Reflexive Side side shared by two triangles

Name That Postulate SAS SAS SSA AAS Vertical Angles Reflexive Property (when possible) Vertical Angles Reflexive Property SAS SAS Vertical Angles Reflexive Property SSA AAS Not enough info!

Reflexive Sides and Angles When two triangles overlap, there may be additional congruent parts. Reflexive Side side shared by two triangles Reflexive Angle angle shared by two

Let’s Practice B  D AC  FE A  F Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: B  D For SAS: AC  FE A  F For AAS:

Problem #1

Step 1: Mark the Given

Step 3: Choose a Method SSS SAS ASA AAS HL

Step 4: List the Parts S S S … in the order of the Method STATEMENTS REASONS S S S … in the order of the Method

Step 5: Fill in the Reasons STATEMENTS REASONS S S S (Why did you mark those parts?)

Step 6: Is there more? S S S The “Prove” Statement is always last ! STATEMENTS REASONS S S S

Problem #2

Step 1: Mark the Given

Step 3: Choose a Method SSS SAS ASA AAS HL

Step 4: List the Parts S A S … in the order of the Method STATEMENTS REASONS S A S … in the order of the Method

Step 5: Fill in the Reasons STATEMENTS REASONS S A S (Why did you mark those parts?)

Step 6: Is there more? S A S The “Prove” Statement is always last ! STATEMENTS REASONS S A S

Problem #3

Step 1: Mark the Given

Step 3: Choose a Method SSS SAS ASA AAS HL

Step 4: List the Parts A S A … in the order of the Method STATEMENTS REASONS A S A … in the order of the Method

Step 6: Is there more? A S A The “Prove” Statement is always last ! STATEMENTS REASONS A S A

Problem #4 AAS Statements Reasons Given Given AAS Postulate Vertical Angles Thm Given AAS Postulate

Problem #5 HL Statements Reasons Given Given Reflexive Property Given ABC, ADC right s, Prove: Statements Reasons Given 1. ABC, ADC right s Given Reflexive Property HL Postulate

Congruence Proofs 1. Mark the Given. 2. Mark … Reflexive Sides or Angles / Vertical Angles Also: mark info implied by given info. 3. Choose a Method. (SSS , SAS, ASA) 4. List the Parts … in the order of the method. 5. Fill in the Reasons … why you marked the parts. 6. Is there more?

Given implies Congruent Parts segments midpoint angles parallel segments segment bisector angles angle bisector angles perpendicular

Example Problem

… and what it implies Step 1: Mark the Given

Reflexive Sides Vertical Angles Step 2: Mark . . . … if they exist.

Step 3: Choose a Method SSS SAS ASA AAS HL

Step 4: List the Parts S A … in the order of the Method STATEMENTS REASONS S A … in the order of the Method

Step 5: Fill in the Reasons STATEMENTS REASONS S A S (Why did you mark those parts?)

Step 6: Is there more? STATEMENTS REASONS S 1. 2. 3. 4. 5. A S

Midpoint implies segments. Back Midpoint implies segments. STATEMENTS REASONS S … 3. 3. Given

Parallel implies angles. Back Parallel implies angles. STATEMENTS REASONS A A

Seg. bisector implies segments. Back Seg. bisector implies segments. STATEMENTS REASONS S … S

Angle bisector implies angles. Back Angle bisector implies angles. STATEMENTS REASONS A …

implies right ( ) angles. Back STATEMENTS REASONS A … S 4. 4. Given

Congruent Triangles Proofs 1. Mark the Given and what it implies. 2. Mark … Reflexive Sides / Vertical Angles 3. Choose a Method. (SSS , SAS, ASA) 4. List the Parts … in the order of the method. 5. Fill in the Reasons … why you marked the parts. 6. Is there more?

Using CPCTC in Proofs According to the definition of congruence, if two triangles are congruent, their corresponding parts (sides and angles) are also congruent. This means that two sides or angles that are not marked as congruent can be proven to be congruent if they are part of two congruent triangles. This reasoning, when used to prove congruence, is abbreviated CPCTC, which stands for Corresponding Parts of Congruent Triangles are Congruent.

Corresponding Parts of Congruent Triangles For example, can you prove that sides AD and BC are congruent in the figure at right? The sides will be congruent if triangle ADM is congruent to triangle BCM. Angles A and B are congruent because they are marked. Sides MA and MB are congruent because they are marked. Angles 1 and 2 are congruent because they are vertical angles. So triangle ADM is congruent to triangle BCM by ASA. This means sides AD and BC are congruent by CPCTC.

Corresponding Parts of Congruent Triangles A two column proof that sides AD and BC are congruent in the figure at right is shown below: Statement Reason MA @ MB Given ÐA @ ÐB Ð1 @ Ð2 Vertical angles DADM @ DBCM ASA AD @ BC CPCTC

Corresponding Parts of Congruent Triangles A two column proof that sides AD and BC are congruent in the figure at right is shown below: Statement Reason MA @ MB Given ÐA @ ÐB Ð1 @ Ð2 Vertical angles DADM @ DBCM ASA AD @ BC CPCTC

Corresponding Parts of Congruent Triangles Sometimes it is necessary to add an auxiliary line in order to complete a proof For example, to prove ÐR @ ÐO in this picture Statement Reason FR @ FO Given RU @ OU UF @ UF reflexive prop. DFRU @ DFOU SSS ÐR @ ÐO CPCTC