Question

8 a tree has a shadow 15 meters long and the height of the tree is unknown. a pole that is 5 meters tall casts a shadow 6 meters long. are the triangles formed by the tree and the pole similar?a no, the sides are not proportionalb yes, by the aa criteriac yes, by the sas criteriad no, the angles are not congruent9 a rectangular billboard frame has two sides measuring 10 meters and 12 meters, with the included angle measuring 60°. another frame has sides of 8 meters and 9 meters with the same 60° included angle. areat a yes, by the sss criteriab no, the sides are not proportionalc yes, by the sas criteriad no, the angles do not match10. two triangles are similar and one triangle has sides 8 cm, 12 cm, and 16 cm. the corresponding sides of the other triangle are proportional, with the shortest side measuring 6 cm. what is the length of the longa 12 cmb 20 cmc 24 cmd 18 cm11 a flagpole has sides that form a triangle with the ground. the height of the flagpole is 8 meters, and the distance to the tip of its shadow is 10 meters, forming a 50° angle with the ground. a nearby pole has a hea no, the angles are not congruentb yes, by the sss criteriac no, the sides are not proportionald yes, by the sas criterion12 when proving similarity using the aa criterion, how many angle pairs need to be equal?a all corresponding angles must be proportionalb 2c 3d 113 which of the following best describes the triangle proportionality theorem?a the sum of the interior angles of a triangle is 180 degreesb if two triangles are similar, their corresponding angles are equalc if two triangles are congruent, their corresponding sides are proportionald a line parallel to one side of a triangle divides the other two sides proportionally14 if two triangles are similar, what must be true about their corresponding angles?a they are complementaryb they are equalc they are supplementaryd they are congruent

Explanation:

Step1: Analyze question 8

• Both the tree - pole and the stick - shadow form right - angled triangles.
• The angle of elevation of the sun is the same for both (common angle).
• By AA (Angle - Angle) criterion (right - angle and common angle), the triangles are similar.

Step2: Analyze question 9

• For the first frame: sides \(a = 10\), \(b = 12\), included angle \(\theta=60^{\circ}\)
• For the second frame: sides \(a'=8\), \(b' = 9\)
• Check the ratio \(\frac{10}{8}=\frac{5}{4}\) and \(\frac{12}{9}=\frac{4}{3}\). Since \(\frac{5}{4}

eq\frac{4}{3}\), sides are not proportional.

Step3: Analyze question 10

• Let the sides of the first triangle be \(a = 8\), \(b = 12\), \(c = 16\)
• The shortest side of the first triangle is \(a = 8\) and of the second triangle is \(a'=6\)
• The ratio of similarity \(k=\frac{6}{8}=\frac{3}{4}\)
• The longest side of the first triangle is \(c = 16\). The longest side of the second triangle \(c'=16\times\frac{3}{4}=12\)

Step4: Analyze question 11

• For the flagpole: height \(h = 8\), hypotenuse \(H = 10\)
• For the nearby pole: height \(h'=12\), hypotenuse \(H'=15\)
• Check the ratio \(\frac{8}{12}=\frac{2}{3}\) and \(\frac{10}{15}=\frac{2}{3}\), and included right - angle. By SAS (Side - Angle - Side) criterion, they are similar.

Step5: Analyze question 12

• In AA (Angle - Angle) criterion for similarity of triangles, 2 pairs of corresponding angles need to be equal.

Step6: Analyze question 13

• The Triangle Proportionality Theorem states that a line parallel to one side of a triangle divides the other two sides proportionally.

Step7: Analyze question 14

• If two triangles are similar, their corresponding angles are equal.

Answer:

1. b. Yes, by the AA Criterion
2. b. No, the sides are not proportional
3. a. 12 cm
4. d. Yes, by the SAS Criterion
5. b. 2
6. d. A line parallel to one side of a triangle divides the other two sides proportionally
7. b. They are equal