Question

adawi geometry
practice 6 - 3
name
date__________ period
state if the triangles in each pair are similar. if so, state how you know they are similar.

1. $\triangle cba - \triangle stu$

a) not similar
b) similar; aa similarity
c) similar; sss similarity
d) similar; sas similarity

1. $\triangle qrs - \triangle edc$

a) similar; sas similarity
b) similar; aa similarity
c) not similar
d) similar; sss similarity

Explanation:

Problem 1: $\triangle CBA$ and $\triangle STU$

Step 1: Identify Angles

In $\triangle CBA$ and $\triangle STU$, we check for equal angles. From the diagram, $\angle A$ in $\triangle CBA$ and $\angle T$ in $\triangle STU$? Wait, no, looking at the triangles: $\triangle CBA$ has angles at $A$, $B$, $C$ and $\triangle STU$ (wait, the labels: $\triangle CBA$ and $\triangle STU$? Wait the lower triangle is $\triangle STU$? Wait the first triangle: $\triangle CBA$ with $\angle A$ (red) and $\angle B$ (red), and the lower triangle $\triangle STU$ (wait, the labels: $S$, $T$, $U$? Wait the lower triangle has vertex $T$ at the top, $S$ and $U$ at the base? Wait, the angles: $\angle A$ in $\triangle CBA$ and $\angle T$? No, wait, the first triangle: $\angle A$ (red) and $\angle B$ (red), and the lower triangle: $\angle T$ (top) and $\angle U$ (base right) red. Wait, actually, $\angle A$ in $\triangle CBA$ and $\angle T$? No, maybe $\angle A$ and $\angle T$? Wait, no, the key is AA (Angle-Angle) similarity: if two angles of one triangle are equal to two angles of another triangle, the triangles are similar.

Looking at $\triangle CBA$: $\angle A$ (red) and $\angle B$ (red). In $\triangle STU$ (lower triangle), $\angle T$ (top) and $\angle U$ (base right) red. So $\angle A = \angle T$ (vertical? No, but the diagram shows two angles equal. So two angles are equal, so by AA similarity, the triangles are similar.

Step 2: Determine Similarity Criterion

Since two angles are equal, the AA (Angle-Angle) similarity criterion applies. So the triangles are similar by AA similarity.

Answer:

b) similar; AA similarity

Problem 2: $\triangle QRS$ and $\triangle EDC$ (Wait, the labels: $\triangle EDC$ and $\triangle QRS$? Wait the left triangle is $\triangle EDC$ with $C$, $E$, $D$: $CE = 8$, $ED = 16$. The right triangle is $\triangle QRS$? Wait no, the right triangle is $\triangle QBE$? Wait the labels: $C$, $E$, $D$ (left triangle: $CE = 8$, $ED = 16$, right angle at $E$? Wait, $\angle E$ in $\triangle EDC$ and $\angle Q$ in $\triangle QRS$ (right triangle) are right angles? Wait, the sides: $CE = 8$, $ED = 16$; $SQ = 18$? Wait no, the right triangle: $S$ to $Q$ is 18? Wait no, the left triangle: $CE = 8$, $ED = 16$; right triangle: $S$ to $Q$ is 18? Wait, no, the sides: $CE = 8$, $ED = 16$; and the right triangle: $SQ = 18$? Wait, no, the right triangle: $S$ to $Q$ is 18? Wait, the sides adjacent to the angle: in $\triangle EDC$, $CE = 8$, $ED = 16$; in the right triangle, let's see: the side adjacent to $\angle E$ (left triangle) is $CE = 8$, and the side adjacent to $\angle Q$ (right triangle) is $SQ = 18$? Wait, no, the sides: $CE = 8$, $ED = 16$; and the right triangle: the side adjacent to $\angle Q$ is $SQ = 18$? Wait, no, the other side: $ED = 16$, and the right triangle's side is $36$? Wait, $16$ and $36$? Wait, $CE = 8$, $SQ = 18$; $ED = 16$, $EB = 36$? Wait, no, the ratio of $CE$ to $SQ$ is $8/18 = 4/9$, and $ED$ to $EB$ is $16/36 = 4/9$. And the included angle: $\angle E$ and $\angle Q$ are equal (both right angles? Wait, the diagram shows $\angle E$ and $\angle Q$ as right angles? Wait, the left triangle: $\angle E$ is a right angle (marked), and the right triangle: $\angle Q$ is a right angle (marked). So we have two sides in proportion and the included angle equal. So the SAS (Side-Angle-Side) similarity criterion: if two sides of one triangle are proportional to two sides of another triangle, and the included angles are equal, the triangles are similar.

Step 1: Check Side Ratios

Calculate the ratio of the sides: $CE/SQ = 8/18 = 4/9$ and $ED/EB = 16/36 = 4/9$. So the sides are proportional.

Step 2: Check Included Angle

The included angle between the sides $CE$ and $ED$ is $\angle E$, and between $SQ$ and $EB$ is $\angle Q$. These angles are equal (both right angles, so $90^\circ$).

Step 3: Apply SAS Similarity

Since two sides are proportional and the included angle is equal, the triangles are similar by SAS similarity.