Question

7 practice 7 (from unit 3, lesson 5)
here is a diagram with triangles abc and aed.
∠aed ≅ ∠abc
m∠ade = x°
select all true statements given that angle aed is congruent to angle abc.
a angle acb is 180 - x°.
b angle acb is x°.
c triangle acb is similar to triangle ade.
d ad = 1/3 ac
e ad = 1/2 dc

Explanation:

Step1: Check similarity of triangles

Since \(\angle AED\cong\angle ABC\) and \(\angle A\) is common to both \(\triangle ADE\) and \(\triangle ACB\). By the AA (Angle - Angle) similarity criterion, \(\triangle ADE\sim\triangle ACB\).

Step2: Use similarity ratio

We have \(\frac{AE}{AB}=\frac{5}{5 + 2}=\frac{5}{10}=\frac{1}{2}\), \(\frac{AD}{AC}\). Let \(AD = y\) and \(DC=z\), then \(AC=y + z\). Since \(\triangle ADE\sim\triangle ACB\), \(\frac{AD}{AC}=\frac{AE}{AB}=\frac{1}{2}\), so \(2AD=AC\), \(2AD=AD + DC\), \(AD = DC\). Also, \(\angle ADE=\angle ACB\) (corresponding angles of similar triangles).

Answer:

B. Angle \(ACB\) is \(x^{\circ}\).
C. Triangle \(ACB\) is similar to triangle \(ADE\).