Question

according to the diagram below, which similarity statements are true? check all that apply. a. △abc ~ △adc b. △abd ~ △bcd c. △abc ~ △adb d. △abc ~ △bdc

Explanation:

Step1: Identify right - angled triangles

In right - angled triangle $\triangle ABC$ with right - angle at $B$ and altitude $BD$ drawn to the hypotenuse $AC$. We use the geometric mean theorem for right - angled triangles.

Step2: Check angle - angle similarity

In $\triangle ABC$ and $\triangle ADB$, $\angle A$ is common and $\angle ABC=\angle ADB = 90^{\circ}$. By the AA (angle - angle) similarity criterion, $\triangle ABC\sim\triangle ADB$.
In $\triangle ABC$ and $\triangle BDC$, $\angle C$ is common and $\angle ABC=\angle BDC = 90^{\circ}$. By the AA similarity criterion, $\triangle ABC\sim\triangle BDC$.
In $\triangle ABD$ and $\triangle BCD$, $\angle ADB=\angle BDC = 90^{\circ}$, and $\angle ABD+\angle A = 90^{\circ}$, $\angle C+\angle A=90^{\circ}$, so $\angle ABD=\angle C$. By the AA similarity criterion, $\triangle ABD\sim\triangle BCD$.

Answer:

B. $\triangle ABD\sim\triangle BCD$
C. $\triangle ABC\sim\triangle ADB$
D. $\triangle ABC\sim\triangle BDC$