Give reasons for each step in this proof. B D A E C Given: C is the midpoint of AC, C is the midpoint of DB Prove: AB is congruent to ED StatementsReasons C is the midpoint of AC, C is the midpoint of DB AC is congruent to CE DC is congruent to CB <ACB is congruent to <DCE Triangle ACB is congruent to triangle ECD AB is congruent to ED Vertical angles are congruent SAS CPCTC Midpoint Theorem Given Midpoint Theorem

Proving Triangles Congruent: AAS

Angle-Angle-Side Theorem (AAS)- If two angles and a nonincluded side of one triangle are congruent to the corresponding two angles and side of a second triangle, the two triangles are congruent.

Isosceles Triangle Theorem- If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Theorem 4-7- If two angles of a triangle are congruent, then the sides opposite those angles are congruent. Corollary 4-3- A triangle is equilateral if and only if it is equiangular. Corollary 4-4- Each angle of an equilateral triangle measures 60 degrees.

Example 1) In isosceles triangle DEF, <D is the vertex angle. If m<E = 2x + 40, and m<F = 3x + 22, find the measure of each angle of the triangle. FE D Since it is isosceles, the base angles are congruent. m<E = m<F 2x + 40 = 3x = x = x Plug 18 in for x in either equation. m<E = 2x + 40 m<E = 2(18) + 40 m<E = m<E = 76 = m<F Now all the angles in a triangle add up to = m<E + m<F + m<D 180 = m<D 180 = m<D 28 = m<D

Example 2) In isosceles triangle ISO with base SO, m<S = 5x - 18 and m<O = 2x Find the measure of each angle of the triangle. OS I Since it is isosceles, the base angles are congruent. m<O = m<S 2x + 21 = 5x = 3x – = 3x 13 = x Plug 13 in for x in either equation. m<O = 2x + 21 m<O = 2(13) + 21 m<O = m<O = 47 = m<S Now all the angles in a triangle add up to = m<O + m<S + m<I 180 = m<I 180 = 94 + m<I 86 = m<I

Example 3) Given: AB congruent to EB and <DEC congruent to <B Prove: Triangle ABE is equilateral. C A B StatementsReasons AB congruent to EB and <DEC congruent to <B <A congruent to <AEB <AEB is congruent to <DEC <A congruent to <AEB congruent to <B Given Isosceles Triangle Theorem Transitive Property of Equality Triangle AEB is equiangularDefinition of equiangular triangles D E Vertical angles are congruent Triangle ABE is equilateral If a Triangle is equiangular, it is equilateral.

Example 4) Find the value of x. All the angles in a triangle add up to = 60 + y + y 180 = y 120 = 2y 60 = y So the top triangle is an equilateral triangle. So all the sides equal 2x + 5. The bottom triangle is an isosceles triangle. 2x + 5 = 3x – 13 5 = x – = x 60 2x + 5 3x - 13 yy

Example 5) Find the value of x. This is an isosceles triangle so two of the angles are equal to 3x + 8. All the angles in a triangle add up to = 2x x x = 8x = 8x 18 = x 2x x + 8

Example 6) In triangle ABD, AB is congruent to BD, m<A is 12 less than 3 times a number, and m<D is 13 more than twice the same number. Find m<B. Start by drawing a diagram. Now since AB is congruent to BD we know Triangle ABD is an isosceles triangle. m<A = m<D 3x – 12 = 2x + 13 x – 12 = 13 x = 25 Plug 25 in for x in either equation. m<A = 3x – 12 m<A = 3(25) – 12 m<A = 75 – 12 m<A = 63 = m<D B D A All the angles in a triangle add up to = m<A + m<B + m<D 180 = 63 + m<B = m<B 54 = m<B

Example 7) Given: AD congruent to AE and <ACD congruent to <ABE Prove: Triangle ABC is isosceles. E StatementsReasons AD congruent to AE and <ACD congruent to <ABE <A is congruent to <A AC is congruent to AB Triangle ACD is congruent to triangle ABE Given AAS CPCTC Congruence of angles is reflexive C A B D Definition of isosceles triangleTriangle ABC is isosceles.

Refer to the figure T S R CF D Example 8) Name the included side for <1 and <4. FD Example 9) Name the included side for <7 and <8. DS Example 10) Name a nonincluded side for <5 and <6. FD, FR, RS, ST, CT, or DC Example11) Name a nonincluded side for <9 and <10. DS, CT, ST, SR, RF, or FD

Refer to the figure T S R CF D Example 12) CT is included between which two angles? <10 and <11 Example 13) In triangle FDR, name a pair of angles so that FR is not included. <2 and <4 or <1 and <4 Example 14) If <1 is congruent to <6, <4 is congruent to <3, and FR is congruent to DS, then triangle FDR is congruent to triangle ______ by ______. SRD; AAS

Refer to the figure T S R CF D Example 15) If <5 is congruent to <7, <6 is congruent to <8, and DS is congruent to DS, then triangle TDS is congruent to triangle ______ by ______. RDS; ASA Example 16) If <4 is congruent to <9, what sides would need to be congruent to show triangle FDR is congruent to triangle CDT? FD congruent to CD and DR congruent to DT