Question

1. which of the following statements is true for similar triangles?

a. corresponding angles are congruent and sides are proportional
b. only one angle is congruent
c. none of the sides are proportional
d. all sides are equal

1. what mathematical property justifies why two triangles are similar when two angles are congruent?

a. transitive property
b. reflexive property
c. angle sum property of a triangle
d. distributive property

1. if △abc ~ △def and ab = 12 cm, bc = 9 cm, and de = 4 cm, what is the length of ef?

a. 6 cm
b. 3 cm
c. 9 cm
d. 12 cm

1. if two triangles have side ratios ½, ½, ½, what is their scale factor?

a. ½
b. 1
c. 3
d. 2

1. when using aa similarity in coordinate geometry, which of the following tools is commonly used?

a. area formula
b. compass
c. slope formula
d. protractor

1. in a real - world scenario, which of the following could be solved using eae similarity?

a. measuring the perimeter of a square
b. finding the area of a circle
c. calculating the volume of a cube
d. comparing the shapes of bird nests in different regions

1. which of the following is used to prove that two triangles are similar using aa similarity?

a. one pair of sides and one pair of angles
b. all angles and all sides
c. two pairs of angles
d. three pairs of sides

Explanation:

1. For the first question:

• Similar triangles have corresponding angles congruent and sides proportional. Option b (only one angle congruent) is not sufficient for similarity. Option c (some sides proportional) is not enough as angles also matter. Option d (all sides equal) is for congruent triangles.

2. For the second question:

• When two angles of a triangle are congruent, the third angle is also congruent (by Angle Sum Property of a Triangle: sum of angles in a triangle is \(180^{\circ}\)). So if two angles are congruent, the triangles are similar.

3. For the third question:

• If \(\triangle ABC\sim\triangle DEF\), then \(\frac{AB}{DE}=\frac{BC}{EF}\). Given \(AB = 12\mathrm{cm}\), \(BC=9\mathrm{cm}\), \(DE = 4\mathrm{cm}\). Substitute into the proportion \(\frac{12}{4}=\frac{9}{EF}\). Cross - multiply: \(12EF=4\times9\), \(12EF = 36\), \(EF=\frac{36}{12}=3\mathrm{cm}\).

4. For the fourth question:

• The scale factor is the ratio of corresponding sides. If the side ratios are \(\frac{1}{3},\frac{1}{3},\frac{1}{3}\), the scale factor is \(\frac{1}{3}\).

5. For the fifth question:

• In coordinate geometry for AA similarity, the slope formula is used to check if angles (by checking if lines are parallel, which gives congruent angles) are congruent. Area formula is for area, compass for drawing circles/arcs, protractor for measuring angles (not typically in coordinate - based AA similarity checks).

6. For the sixth question:

• EAE (which is similar to AA) similarity is about shape (angle - angle relationships). Comparing the shapes of bread crusts in different regions (which is about angle - related shape) can use EAE. Measuring perimeter of a square, finding area of a circle, calculating volume of a cube are about other geometric properties (not angle - based similarity).

7. For the seventh question:

• AA (Angle - Angle) similarity requires two pairs of congruent angles. One pair of sides and one pair of angles (not enough for AA), all angles and all sides (for congruence), three pairs of sides (SSS similarity) are not for AA.

Answer:

1. b
2. c
3. b
4. a
5. c
6. d
7. c