Converse of the Isosceles Triangle Theorem

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A B C S R T C A C CONVERSE OF THE ISOSCELES TRIANGLE THEOREM Illustrative Example: Prove that an equiangular triangle ABC is also equilateral. Proof: Statement Reason 1. ∠ A ≅ ∠ C 1. Definition of an equiangular triangle 2. AB ≅BC 2. Converse of the Isosceles Triangle Theorem 3. ∠ A ≅ ∠ B 3. Definition of an equiangular triangle 4. AC ≅ BC 4. Converse of the Isosceles Triangle Theorem 5. AC ≅ BC ≅ AC 5. Transitive Property 6. ∆ ABCis equilateral. 6. Definition of an equilateral triangle. Given: ∆ RST, with ∠ S ≅ ∠ T Prove: SR ≅ TR Proof: Statement Reason 1. Draw RA , the angle bisector of R that intersects ST at A 1. Every angle has a unique ray that bisects it. 2. ∠ ARS ≅ ∠ ART 2. Definition of angle bisector 3. RA ≅ RA 3. Reflexive Property 4. ∠ S ≅ ∠ T 4. Given

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GEOMETRY

Transcript of Converse of the Isosceles Triangle Theorem

Page 1: Converse of the Isosceles Triangle Theorem

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CONVERSE OF THE ISOSCELES TRIANGLE THEOREM

Illustrative Example:Prove that an equiangular triangle ABC is also equilateral.

Proof:Statement Reason

1. ∠A ≅ ∠C 1. Definition of an equiangular triangle2. AB ≅ BC 2. Converse of the Isosceles Triangle

Theorem3. ∠A ≅ ∠B 3. Definition of an equiangular triangle4. AC ≅ BC 4. Converse of the Isosceles Triangle

Theorem5. AC ≅ BC ≅ AC 5. Transitive Property6. ∆ ABC is equilateral. 6. Definition of an equilateral triangle.

Given: ∆RST, with ∠S ≅ ∠TProve: SR ≅TR

Proof:Statement Reason

1. Draw R⃗A , the angle bisector of R that intersects ST at A

1. Every angle has a unique ray that bisects it.

2. ∠ARS ≅ ∠ART 2. Definition of angle bisector3. RA ≅ RA 3. Reflexive Property4. ∠S ≅ ∠T 4. Given5. ∆RSA ≅ ∆RTA 5. SAA Congruence Postulate6. SR ≅TR 6. Corresponding parts of congruent

triangles are congruent.

Page 2: Converse of the Isosceles Triangle Theorem

ACTIVITY1. Draw a scalene triangle. Name it as ∆BAC.2. Find the length of the segments.

BC = _____________AC = _____________

3. What is the relationship between BC and ACBC __________ AC

4. Find:m∠A = ___________m∠B = ___________

5. What is the relationship betweenm∠A ______________ m∠B ?

6. What conclusion can you formulate? ____________________________________________________________________________________________________________________________

POSSIBLE ANSWERS:2. BC = 4 cm, AC = 6 cm

3. This means that BC ≇ AC.

4. m∠A = 42, m∠B = 88

5. m∠A ≇m∠B

6. If the sides of a triangle are not congruent, then the angles opposite these sides are not congruent.

Page 3: Converse of the Isosceles Triangle Theorem

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INEQUALITIES IN A TRIANGLE

Theorem:If two sides of a triangle are not congruent, then the angles opposite these two sides are not

congruent, and the larger angle is opposite the longer side.

Proof:

Given: RS > STProve: ∠SRT < ∠STR

Statement Reason1. Let U be a point on ST such that SU = RS 1. By construction2. SU ≅ RS 2. Definition of Congruent Segments3. ∠SRU ≅ ∠U 3. In the same triangle, angles opposite

congruent sides are congruent.4. RS > ST 4. Given5. SU > ST 5. Substitution6. m∠SRU = ∠SRT + ∠TRU 6. Angle Addition Postulate7. m∠SRT < ∠SRU 7. Definition of Inequality8. ∠SRT < ∠U 8. Substitution9. ∠U < ∠STR 9. Exterior Angle Theorem10. ∠SRT < ∠STR 10. Transitivity

Exterior Angle Theorem:For any triangle, the measure of an exterior angle is equal to the sum of the measures of its

two remote interior angles.

Theorem:If two angles of a triangle are not congruent, then the sides opposite these two angles are not

congruent, and the longer side is opposite the larger angle.

Page 4: Converse of the Isosceles Triangle Theorem

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