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Download Theorem 20: If two sides of a triangle are congruent, the angles
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Transcript
Theorem 20: If two sides of a triangle are congruent, the angles opposite the sides are congruent. ( If , then . ) Theorem 20: If two sides of a triangle are congruent, the angles opposite the sides are congruent. ( If , then . ) Given: AB ~ = AC Prove: A B~ = C B Proof Statement C Reason Theorem 20: If two sides of a triangle are congruent, the angles opposite the sides are congruent. ( If , then . ) Given: AB ~ = AC Prove: A B~ = C B Proof Statement C Reason Theorem 20: If two sides of a triangle are congruent, the angles opposite the sides are congruent. ( If , then . ) Given: AB ~ = AC Prove: A A B~ = C B Proof Statement C C Reason B Theorem 20: If two sides of a triangle are congruent, the angles opposite the sides are congruent. ( If , then . ) Given: AB ~ = AC Prove: A A B~ = C B Proof Statement C C Reason B Theorem 20: If two sides of a triangle are congruent, the angles opposite the sides are congruent. ( If , then . ) Given: AB ~ = AC Prove: A A B~ = C B Proof Statement 1. AB ~ = AC C C Reason 1. Given B Theorem 20: If two sides of a triangle are congruent, the angles opposite the sides are congruent. ( If , then . ) Given: AB ~ = AC Prove: A A B~ = C B Proof Statement 1. AB ~ = AC 2. BC ~ = BC C C Reason 1. Given 2. Ref. B Theorem 20: If two sides of a triangle are congruent, the angles opposite the sides are congruent. ( If , then . ) Given: AB ~ = AC Prove: A A B~ = C B Proof Statement 1. AB ~ = AC 2. BC ~ = BC 3. ABC ~ = ACB C C Reason 1. Given 2. Ref. 3. SSS B Theorem 20: If two sides of a triangle are congruent, the angles opposite the sides are congruent. ( If , then . ) Given: AB ~ = AC Prove: A A B~ = C B Proof Statement 1. AB ~ = AC 2. BC ~ = BC 3. ABC ~ = ACB 4. B ~ = C C C Reason 1. Given 2. Ref. 3. SSS 4. CPCTC B Theorem 21: If two angles of a triangle are congruent, the sides opposite the angles are congruent. ( If , then . ) Theorem 21: If two angles of a triangle are congruent, the sides opposite the angles are congruent. (If , then .) ~ = C Prove: AB ~ = AC Given: A B B Proof Statement C Reason Theorem 21: If two angles of a triangle are congruent, the sides opposite the angles are congruent. (If , then .) ~ = C Prove: AB ~ = AC Given: A B B Proof Statement 1. B ~ = C 2. BC ~ = BC 3. ABC ~ = ACB 4. AB ~ = AC A C C Reason 1. Given 2. Ref. 3. ASA 4. CPCTC B The two theorems tell us: If at least two sides of a triangle are congruent, the triangle is isosceles. If at least two angles of a triangle are congruent, the triangle is isosceles. Theorem 20: If two sides of a triangle are congruent, the angles opposite the sides are congruent. ( If , then . ) The inverse of Theorem 20 is true: If two sides of a triangle are not congruent, then the angles opposite them are not congruent, and the larger angle is opposite the longer side. ( If , then . ) Theorem 21: If two angles of a triangle are congruent, the sides opposite the angles are congruent. (If , then .) The inverse of Theorem 21 is true: If two angles of a triangle are not congruent, then the sides opposite them are not congruent, and the longer side is opposite the larger angle. ( If , then . ) The median to the base of an isosceles triangle bisects the vertex angle. The median to the base of an isosceles triangle bisects the vertex angle. A Given: Isosceles ABC with vertex A and median AD Prove: Proof BAD ~ = CAD Statement B Reason D C The median to the base of an isosceles triangle bisects the vertex angle. A Given: Isosceles ABC with vertex A and median AD Prove: Proof BAD ~ = CAD Statement B Reason D C The median to the base of an isosceles triangle bisects the vertex angle. A Given: Isosceles ABC with vertex A and median AD Prove: Proof BAD ~ = CAD Statement B Reason D C The median to the base of an isosceles triangle bisects the vertex angle. A Given: Isosceles ABC with vertex A and median AD Prove: Proof BAD ~ = CAD Statement B Reason D C The median to the base of an isosceles triangle bisects the vertex angle. A Given: Isosceles ABC with vertex A and median AD Prove: Proof BAD ~ = CAD Statement 1. Isosceles ABC with vertex A B Reason 1. Given D C The median to the base of an isosceles triangle bisects the vertex angle. A Given: Isosceles ABC with vertex A and median AD Prove: Proof BAD ~ = CAD Statement 1. Isosceles ABC with vertex A S 2. AB ~ = AC B D C Reason 1. Given 2. Legs of isos. are ~ =. The median to the base of an isosceles triangle bisects the vertex angle. A Given: Isosceles ABC with vertex A and median AD Prove: Proof BAD ~ = CAD Statement 1. Isosceles ABC with vertex A S 2. AB ~ = AC A 3. B ~ = C B D C Reason 1. Given 2. Legs of isos. 3. If , then . are ~ =. The median to the base of an isosceles triangle bisects the vertex angle. A Given: Isosceles ABC with vertex A and median AD Prove: Proof BAD ~ = CAD Statement 1. Isosceles ABC with vertex A S 2. AB ~ = AC A 3. B ~ = C 4. AD is a median B D C Reason 1. Given 2. Legs of isos. 3. If , then . 4. Given are ~ =. The median to the base of an isosceles triangle bisects the vertex angle. A Given: Isosceles ABC with vertex A and median AD Prove: Proof BAD ~ = CAD Statement 1. Isosceles ABC with vertex A S 2. AB ~ = AC A 3. B ~ = C 4. AD is a median 5. D is mdpnt. of AD B D C Reason 1. Given 2. Legs of isos. are ~ =. 3. If , then . 4. Given 5. Def. of median The median to the base of an isosceles triangle bisects the vertex angle. A Given: Isosceles ABC with vertex A and median AD Prove: Proof BAD ~ = CAD Statement 1. Isosceles ABC with vertex A S 2. AB ~ = AC A 3. B ~ = C 4. AD is a median 5. D is mdpnt. of AD S 6. BD ~ = CD B D C Reason 1. Given 2. Legs of isos. are ~ =. 3. If , then . 4. Given 5. Def. of median 6. Def. of midpoint The median to the base of an isosceles triangle bisects the vertex angle. A Given: Isosceles ABC with vertex A and median AD Prove: Proof BAD ~ = CAD Statement 1. Isosceles ABC with vertex A S 2. AB ~ = AC A 3. B ~ = C 4. AD is a median 5. D is mdpnt. of AD S 6. BD ~ = CD 7. ABD ~ = ACD B D C Reason 1. Given 2. Legs of isos. are ~ =. 3. If , then . 4. Given 5. Def. of median 6. Def. of midpoint 7. SAS The median to the base of an isosceles triangle bisects the vertex angle. A Given: Isosceles ABC with vertex A and median AD Prove: Proof BAD ~ = CAD Statement 1. Isosceles ABC with vertex A S 2. AB ~ = AC A 3. B ~ = C 4. AD is a median 5. D is mdpnt. of AD S 6. BD ~ = CD 7. ABD ~ = ACD 8. BAD ~ = CAD B D C Reason 1. Given 2. Legs of isos. are ~ =. 3. If , then . 4. Given 5. Def. of median 6. Def. of midpoint 7. SAS 8. CPCTC Theorem 20: If two sides of a triangle are congruent, the angles opposite the sides are congruent. ( If , then . ) Theorem 20: If two sides of a triangle are congruent, the angles opposite the sides are congruent. ( If , then . ) Theorem 21: If two angles of a triangle are congruent, the sides opposite the angles are congruent. ( If , then . )
