Question

using the aa similarity postulate determine whether the triangles are similar. if they are similar, write a similarity statement.
7.
8.
9.
10.
11.
12.

Explanation:

Step1: Recall AA similarity postulate

Two triangles are similar if two angles of one triangle are congruent to two angles of another triangle.

Step2: Analyze triangle 7

In \(\triangle ABC\), angles are \(32^{\circ}\) and another non - given angle. In \(\triangle DEF\), angles are \(33^{\circ}\) and another non - given angle. Since \(32^{\circ}
eq33^{\circ}\) and no other equal angles are indicated, \(\triangle ABC\) and \(\triangle DEF\) are not similar.

Step3: Analyze triangle 8

In \(\triangle NPQ\), angles are \(30^{\circ}\) and \(50^{\circ}\), so the third angle is \(180-(30 + 50)=100^{\circ}\). In \(\triangle RST\), angles are \(30^{\circ}\) and \(110^{\circ}\), so the third angle is \(180-(30+110) = 40^{\circ}\). Since two angles are \(30^{\circ}\) in both, \(\triangle NPQ\sim\triangle RST\) (by AA similarity). The similarity statement is \(\triangle NPQ\sim\triangle RST\).

Step4: Analyze triangle 9

In \(\triangle ABC\), angles are \(47^{\circ}\) and \(92^{\circ}\), so the third angle is \(180-(47 + 92)=41^{\circ}\). In \(\triangle DEF\), angles are \(41^{\circ}\) and \(92^{\circ}\). Since two angles (\(41^{\circ}\) and \(92^{\circ}\)) are congruent in both, \(\triangle ABC\sim\triangle DEF\) (by AA similarity). The similarity statement is \(\triangle ABC\sim\triangle DEF\).

Step5: Analyze triangle 10

In \(\triangle GJH\), one angle is \(58^{\circ}\) and it's a right - triangle, so the other non - right angle is \(180-(90 + 58)=32^{\circ}\). In \(\triangle KML\), one angle is \(32^{\circ}\) and it's a right - triangle. Since two angles (a right - angle and \(32^{\circ}\)) are congruent, \(\triangle GJH\sim\triangle KML\) (by AA similarity). The similarity statement is \(\triangle GJH\sim\triangle KML\).

Step6: Analyze triangle 11

In \(\triangle XYZ\), angles are \(48^{\circ}\) and \(77^{\circ}\), so the third angle is \(180-(48 + 77)=55^{\circ}\). In \(\triangle HGF\), angles are \(55^{\circ}\) and \(48^{\circ}\). Since two angles (\(48^{\circ}\) and \(55^{\circ}\)) are congruent in both, \(\triangle XYZ\sim\triangle HGF\) (by AA similarity). The similarity statement is \(\triangle XYZ\sim\triangle HGF\).

Step7: Analyze triangle 12

In \(\triangle NML\), angles are \(65^{\circ}\) and another \(65^{\circ}\). In \(\triangle KJL\), angles are \(65^{\circ}\) and another \(65^{\circ}\). Since two angles (\(65^{\circ}\) and \(65^{\circ}\)) are congruent in both, \(\triangle NML\sim\triangle KJL\) (by AA similarity). The similarity statement is \(\triangle NML\sim\triangle KJL\).

Answer:

1. Not similar
2. \(\triangle NPQ\sim\triangle RST\)
3. \(\triangle ABC\sim\triangle DEF\)
4. \(\triangle GJH\sim\triangle KML\)
5. \(\triangle XYZ\sim\triangle HGF\)
6. \(\triangle NML\sim\triangle KJL\)