4.3 Analyzing Triangle Congruence Objectives: Identify & use the SSS, SAS, and ASA Congruence Postulates and the AAS and HL Congruence Theorems. Use counterexamples to prove that other side and angle combinations cannot be used to prove triangle congruence. Warm-Up: Which pair of triangles could use the ASA to prove congruency?

Valid or Not?: AAA combination – three angles AAS combination – two angles and a side that is not between them SSA combination- two sides and an angle that is not between them (that is, an angle opposite one of the two sides).

Valid or Not?: AAA combination – three angles you can rule out the AAA combination by finding a counterexample for the following conjecture: Conjecture: If the three angles of one triangle are congruent to the three angles of another triangle, then the triangles are congruent. Counterexample: In the triangles below there are three pairs of congruent angles, but the two triangles are not congruent. Therefore the conjecture is false.

Valid or Not?: SSA combination- two sides & an angle that is not between them you can rule out the SSA combination by finding a counterexample for the following conjecture: Conjecture: If two sides and an angle that is not in between them of one triangle are congruent to two sides and an angle that is not in between them another triangle, then the triangles are congruent. Counterexample: In the triangles below the sides and angles are congruent, but the triangles are not congruent.

Valid or Not?: AAS combination – two angles & a side that is not between them The AAS combination can be converted to the ASA combination: The triangles below are an AAS combination. To convert this to an ASA combination, find the measures of the third angle in each triangle. A D 𝟕𝟓 𝟎 𝟕𝟓 𝟎 𝟔𝟎 𝟎 B C 𝟔𝟎 𝟎 E F A D 𝟔𝟎 𝟎 B 𝟒𝟓 𝟎 C 𝟔𝟎 𝟎 E 𝟒𝟓 𝟎 F

Valid or Not?: The two triangles below represent a version of the AAS, but the triangles are not congruent. There is an important difference between the two triangles. What is it? A D 8 8 𝟒𝟓 𝟎 𝟔𝟎 𝟎 C B 𝟔𝟎 𝟎 E 𝟒𝟓 𝟎 F

AAS (Angle-Angle-Side) Congruence Theorem: For an AAS combination to be used, the congruent parts must correspond. Notice carefully the wording of the following theorem: AAS (Angle-Angle-Side) Congruence Theorem: If two angles and a non-included side of one triangle are congruent to the corresponding angles and non-included side of another triangle, then the triangles are congruent.

Example: Which pairs of congruent triangles can be proven to be congruent by the AAS Congruence Theorem?

HL (Hypotenuse-Leg) Congruence Theorem If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the two triangles are congruent.

∆ABC; AB=6, m<B= 𝟕𝟎 𝟎 , and m<A= 𝟒𝟎 𝟎 Examples: Determine whether the given combination of angles and sides determined a unique triangle. If so, identify the theorem or postulate that supports your answer. ∆ABC; AB=6, m<B= 𝟕𝟎 𝟎 , and m<A= 𝟒𝟎 𝟎 ∆DEF; DE=5, EF=7, and m<F= 𝟑𝟎 𝟎 ∆JKL; m<J= 𝟓𝟎 𝟎 , m<K= 𝟕𝟓 𝟎 and m<L= 𝟓𝟓 𝟎 ∆MNO; MN=8, MO=10, and m<N= 𝟗𝟎 𝟎 ∆PQR; PQ=12, m<P= 𝟒𝟓 𝟎 , and m<R= 𝟏𝟎𝟎 𝟎

Homework: 4.3 Practice Worksheet Textbook pages 230-232 Numbers 6-8, 10 -19

4.3 Triangle Congruence Proofs Objectives: Use definitions & postulates to prove a pair of triangles are congruent. Warm-Up: Are the lines Parallel or curved?

Given: Match the Corresponding Parts AB ≅ DE AC ≅ DF <A ≅ <D Therefore ∆BAC ≅ ∆EDF <A <B <C AB BC AC <D DE EF <E <F DF E

Given: Prove: YW bisects <XYZ XY ≅ YZ ∆YWX ≅ ∆YWZ Y Z X W 1 2 XY ≅ YZ Prove: ∆YWX ≅ ∆YWZ 3 5 4 6 Z X W STATEMENTS REASONS 1. 1. 2. 2. <1 ≅ <2 3. 3. YW ≅ YW 4. ∆YWX ≅ ∆YWZ 4.

Given: Prove: AB || ED CE ≅ CB ∆ABC ≅ ∆DEC A B C STATEMENTS REASONS D 4 6 CE ≅ CB Prove: ∆ABC ≅ ∆DEC 5 C 2 STATEMENTS REASONS 1 3 E D 1. 1. 2. If lines are || then alt int <‘s are ≅ 2. 3. Vertical <‘s are ≅ 3. 4. ∆ABC ≅ ∆DEC 4.

Given: Prove: HJ bisects IK HJ ⊥ IK ∆HJI ≅ ∆HJK H STATEMENTS REASONS K 6 5 HJ ⊥ IK Prove: ∆HJI ≅ ∆HJK 1 2 3 4 STATEMENTS REASONS K I J 1. 1. 2. Def of ⊥ 2. 3. All right <‘s are ≅ 3. 4. J is the midpoint of IK 4. 5. IJ ≅ JK 5. 6. HJ ≅ HJ 6. 7. ∆HJI ≅ ∆HJK 7.

Given: Prove: AB || DC AB ≅ DC ∆ADC ≅ ∆CBA A B D C STATEMENTS REASONS 3 4 2 AB ≅ DC Prove: ∆ADC ≅ ∆CBA 5 1 6 D C STATEMENTS REASONS 1. 1. 2. If lines are || then alt int <‘s are ≅ 2. <3 ≅ <6 3. ref. 3. 4. ∆ADC ≅ ∆CBA 4.

Given: Prove: ME = TA <M ≅ <T ∆MEN ≅ ∆TAN M A N S E STATEMENTS REASONS T 1. 1. 2. 2. 3. 3. 4. 4.

Given: Prove: ∆QRS ≅ ∆TUS QU & RT bisect eachother Q T S STATEMENTS 1 5 Prove: ∆QRS ≅ ∆TUS S 3 4 2 6 STATEMENTS REASONS R U 1. 1. 2. 2. 3. 3. 4. 4.

Homework: Practice Worksheet