Question

which of the following statements is true for similar triangles?

a. none of the sides are proportional

b. only one angle is congruent

c. corresponding angles are congruent and sides are proportional

d. all sides are equal

in triangle pqr and triangle stu, if ∠p = ∠s and pq/st = pr/su, what can be concluded?

a. triangles pqr and stu are congruent by sas congruence

b. triangles pqr and stu are not similar

c. triangles pqr and stu are similar by sas similarity

d. triangles pqr and stu are similar by aaa similarity

which of the following is an example of a correct proportion for similar triangles?

a. ab + de = ac + df

b. ab × de = ac × df

c. ab/de = ac/df

d. ab = ac = de = df

in △ghi and △jkl, if ∠g ≅ ∠j, gh = 10, gi = 15, and jk = 5, what is the missing side jl?

a. 10 cm

b. 7.5 cm

c. 8.5 cm

d. 7 cm

if two triangles have one pair of congruent angles and the surrounding sides are in a ratio of 4, are the triangles similar?

a. no, they must have two pairs of congruent angles

b. no, they must have three pairs of congruent angles

c. yes, by sss similarity

d. yes, by sas similarity

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

a. finding the area of a circle

b. calculating the volume of a cube

c. comparing the shapes of bird nests in different regions

d. measuring the perimeter of a square

Explanation:

Step1: Recall similar triangles properties

Similar triangles have corresponding angles congruent and sides proportional.

Step2: Analyze each option

• Option a: Incorrect, sides are proportional.
• Option b: Incorrect, all corresponding angles are congruent.
• Option c: Correct, matches the definition.
• Option d: Incorrect, that's for congruent triangles.

Step3: Recall SAS 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.

• In the second question, $\angle P=\angle S$ (included angle) and $\frac{PQ}{ST}=\frac{PR}{SU}$ (sides), so by SAS similarity.
• Option a: Congruent is wrong (no info about side - side equality).
• Option b: Incorrect as per SAS similarity.
• Option c: Correct.
• Option d: AAA similarity requires three angles, not applicable here.

Step4: Recall proportion for similar triangles

For similar triangles, the ratio of corresponding sides is equal.

• Option a: Addition is wrong (should be ratio).
• Option b: Multiplication is wrong (should be ratio).
• Option c: $\frac{AB}{DE}=\frac{AC}{DF}$ (correct ratio of corresponding sides).
• Option d: Equality of all sides is for congruent, not similar.

Step5: Use proportion for similar triangles

Since $\triangle GHI\sim\triangle JKL$ (by AA similarity as $\angle G\cong\angle J$), $\frac{GH}{JK}=\frac{GI}{JL}$.
Given $GH = 10$, $GI=15$, $JK = 5$.
$\frac{10}{5}=\frac{15}{JL}$
Cross - multiply: $10\times JL=15\times5$
$JL=\frac{15\times5}{10}=7.5$

Step6: Recall SAS similarity

If one pair of congruent angles and the ratio of surrounding sides (SAS similarity).

• Option a: Incorrect, SAS similarity only needs one pair of congruent angles (included) and side - ratio.
• Option b: Incorrect.
• Option c: SSS similarity is for three sides ratio.
• Option d: Correct (SAS similarity).

Step7: Recall SAS similarity application

SAS similarity is about shape (angle - side - angle proportion).

• Option a: Area of circle is $\pi r^{2}$ (no similarity).
• Option b: Volume of cube is $s^{3}$ (no similarity).
• Option c: Comparing shapes (using similarity, and SAS similarity can be used if angles and side - ratios are considered for the nest shapes).
• Option d: Perimeter of square is $4s$ (no similarity).

Answer:

1. c. Corresponding angles are congruent and sides are proportional
2. c. Triangles PQR and STU are similar by SAS similarity
3. c. $AB/DE = AC/DF$
4. b. $7.5$ cm
5. d. Yes, by SAS similarity
6. c. Comparing the shapes of bird nests in different regions