Question

problems 1 - 3: here are some angle measures for different pairs of triangles. determine whether each pair is similar, not similar, or if there is not enough information to know.

1. triangle a: 53°, 71°

triangle b: 53°, 71°

1. triangle e: 63°, 45°

triangle f: 14°, 71°

1. triangle g: 100°

triangle h: 70°

1. draw two equilateral triangles to fit each category. if its not possible, write

ot possible.\
not congruent
not similar

1. do you think two equilateral triangles will be always, sometimes, or never similar? explain your thinking.
2. in this figure, segment bc is parallel to segment de. how do you know that triangle abc is similar to triangle ade?

Explanation:

Step1: Recall the Angle - Angle (AA) similarity criterion

If two angles of one triangle are equal to two angles of another triangle, then the two triangles are similar.

Step2: Analyze Problem 1

For Triangle \(A\) and Triangle \(B\), two angles \(53^{\circ}\) and \(71^{\circ}\) are equal. By AA similarity criterion, they are similar.

Step3: Analyze Problem 2

For Triangle \(E\): Third angle \(=180-(63 + 45)=72^{\circ}\). For Triangle \(F\): Third angle \(=180-(14 + 71)=95^{\circ}\). No two - angle pairs are equal. So, they are not similar.

Step4: Analyze Problem 3

For Triangle \(G\): Third angle \(=180 - 100=80^{\circ}\). For Triangle \(H\): Third angle \(=180 - 70=110^{\circ}\). No two - angle pairs are equal. So, they are not similar.

Step5: Analyze Problem 4

• Not Congruent: Equilateral triangles have all angles \(60^{\circ}\). We can draw two equilateral triangles with different side lengths (e.g., one with side length \(2\) and another with side length \(3\)). They are similar (by AA, since all angles are \(60^{\circ}\)) but not congruent.
• Not Similar: Since all angles of an equilateral triangle are \(60^{\circ}\), by AA similarity criterion, any two equilateral triangles have three pairs of equal angles. So, it is not possible to draw two non - similar equilateral triangles.

Step6: Analyze Problem 5

All angles of an equilateral triangle are \(60^{\circ}\). For any two equilateral triangles \(\triangle T_1\) and \(\triangle T_2\), \(\angle1=\angle2 = 60^{\circ}\) and \(\angle3=\angle4 = 60^{\circ}\). By AA similarity criterion, two equilateral triangles are always similar.

Step7: Analyze Problem 6

Since \(BC\parallel DE\), \(\angle ABC=\angle ADE\) (corresponding angles) and \(\angle ACB=\angle AED\) (corresponding angles). By AA similarity criterion, \(\triangle ABC\sim\triangle ADE\).

Answer:

1. Similar
2. Not similar
3. Not similar
4. Not Congruent: (Draw two equilateral triangles with different side lengths); Not Similar: Not possible
5. Always similar. Because all angles of equilateral triangles are \(60^{\circ}\), and by AA similarity criterion, two equilateral triangles have two (in fact, three) pairs of equal angles.
6. Since \(BC\parallel DE\), \(\angle ABC=\angle ADE\) and \(\angle ACB=\angle AED\) (corresponding angles). By AA similarity criterion, \(\triangle ABC\sim\triangle ADE\)