Question

1. quadrilaterals abcd and efgh are similar trapezoids. the measure of \\(\overline{ab}\\) is 44 inches, the measure of \\(\overline{bc}\\) is 48 inches, and the measure of \\(\overline{ef}\\) is 33 inches. what is the measure of \\(\overline{fg}\\)? \\(\boldsymbol{\text{a}}\\) 36 inches \\(\boldsymbol{\text{b}}\\) 37 inches \\(\boldsymbol{\text{c}}\\) 38 inches \\(\boldsymbol{\text{d}}\\) 48 inches 2. which similarity theorem can be used to prove the triangles shown are similar? \\(\boldsymbol{\text{a}}\\) angle-angle \\(\boldsymbol{\text{b}}\\) side-side \\(\boldsymbol{\text{c}}\\) side-angle-side \\(\boldsymbol{\text{d}}\\) side-side-side

Explanation:

Problem 1:

Step1: Recall similar trapezoid ratios

For similar trapezoids \(ABCD\) and \(EFGH\), corresponding sides are proportional. So \(\frac{AB}{EF}=\frac{BC}{FG}\).

Step2: Substitute known values

Given \(AB = 44\), \(EF = 33\), \(BC = 48\). Let \(FG = x\). Then \(\frac{44}{33}=\frac{48}{x}\). Simplify \(\frac{44}{33}=\frac{4}{3}\), so \(\frac{4}{3}=\frac{48}{x}\). Cross - multiply: \(4x=48\times3\), \(4x = 144\), \(x=\frac{144}{4}=36\).

Step1: Check angle and side ratios

First, check the angles: both triangles have an \(85^{\circ}\) angle (common angle). Now check the sides around the angle. For the first triangle, sides around \(85^{\circ}\) are \(72\) and \(60\); for the second, \(66\) and \(55\). Calculate ratios: \(\frac{72}{66}=\frac{12}{11}\), \(\frac{60}{55}=\frac{12}{11}\). So the sides around the equal angle are proportional.

Step2: Identify similarity theorem

The Side - Angle - Side (SAS) similarity theorem states that if two sides of one triangle are proportional to two sides of another triangle and the included angles are equal, the triangles are similar. Here, included angle is \(85^{\circ}\) (equal) and sides around it are proportional (\(\frac{72}{66}=\frac{60}{55}=\frac{12}{11}\)). So the theorem is Side - Angle - Side (SAS), which is option C.

Answer:

A. 36 inches

Problem 2: