Question

example 3 (lesson 8-3 practice #9): identify the similar triangles. then find the measure of ( ac ).

example 3 check (lesson 8-3 practice #10): identify the similar triangles. then find the measure of ( jl ).

practice: #s 1-13

summary:

1. if two ______________________ of one triangle are congruent to two

_______________ of another triangle, then the triangles are similar.

Explanation:

First Problem (Lesson 8-3 Practice #9):

Step1: Identify similar triangles

Triangles $\triangle ABC \sim \triangle EBD$ (right triangles, shared angle $\angle B$)

Step2: Set up proportion

Corresponding sides are proportional: $\frac{AC}{ED} = \frac{BC}{BD}$
Substitute values: $\frac{x+1}{x+5} = \frac{12}{15}$

Step3: Simplify & solve for x

Simplify $\frac{12}{15}=\frac{4}{5}$, cross-multiply:
$5(x+1)=4(x+5)$
$5x+5=4x+20$
$5x-4x=20-5$
$x=15$

Step4: Calculate AC length

Substitute $x=15$ into $AC=x+1$:
$AC=15+1=16$

Step1: Identify similar triangles

Triangles $\triangle JKL \sim \triangle NML$ (vertical angles $\angle JLK=\angle NLM$, alternate interior angles are congruent)

Step2: Set up proportion

Corresponding sides are proportional: $\frac{JK}{NM} = \frac{JL}{ML}$
Substitute values: $\frac{16}{4} = \frac{x+18}{x-3}$

Step3: Simplify & solve for x

Simplify $\frac{16}{4}=4$, cross-multiply:
$4(x-3)=x+18$
$4x-12=x+18$
$4x-x=18+12$
$3x=30$
$x=10$

Step4: Calculate JL length

Substitute $x=10$ into $JL=x+18$:
$JL=10+18=28$

This is a key similarity postulate for triangles: if two angles of one triangle match two angles of another, the triangles are similar by AA (Angle-Angle) Similarity.

Answer:

$AC=16$

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Second Problem (Lesson 8-3 Practice #10):