Practice Quiz Triangles

1 If PRQ is an isosceles triangle with PQ = PR, find the measure of QPX. ∆PRQ: Isosceles Triangle The base angles are equal. QPX: Exterior Angle 60 ?

? In the figure, if ABC is the same size and shape as ABD, 2 In the figure, if ABC is the same size and shape as ABD, then the degree measure of BAD = 75 ?

A B In the figure, if side RS = ST and x = 115°, 3 In the figure, if side RS = ST and x = 115°, what is the measure of angle w? Base Angles A = B Isosceles Triangle A B

Supplementary Angles of base angles are equal. 3 In the figure, if side RS = ST and x = 115°, what is the measure of angle w? T = 115 Supplementary Angles of base angles are equal. Base Angles A = B A B w and T are Vertical Angles 115

2z = 70 + z – z – z z = 70 70 2z In the figure, if x = 2z and y = 70, what is the value of z? 4 70 2z 2z = 70 + z Exterior Angles Rule – z – z z = 70

? = 180° – 145° ? = 35° ? In the figure, if side AB = AC and w = 145°, what is the measure of x? ? 145° ? = 180° – 145° ? = 35°

x = 35° + 35° x = 70° 35° 35° In the figure, if side AB = AC and w = 145°, what is the measure of x? Isosceles Triangle 35° 35° 145° Exterior Angles Rule x = 35° + 35° x = 70°

6 If the ratio of the angles of a triangle is 2:3:4, what is the degree measure of the largest angle? Largest Angle 4x 4(20) = 80

B Ratio Vertex : Base : Base 1 : 4 : 4 A C In an isosceles triangle, if the ratio of the vertex angle to the base angle is 1:4, what is the degree measure of the base angle? 7 B Vertex Angle: B Base Angle: A Base Angle: C Ratio Vertex : Base : Base 1 : 4 : 4 A C

B Ratio Vertex : Base : Base 1 : 4 : 4 1x + 4x + 4x = 180 9x = 180 In an isosceles triangle, if the ratio of the vertex angle to the base angle is 1:4, what is the degree measure of the base angle? 7 B Ratio Vertex : Base : Base 1 : 4 : 4 1x + 4x + 4x = 180 9x = 180 x = 20 A = 4(20) = 80 A C

8 The unequal sides of a triangle are integers. If the size order is 5, x, and 15, what is the largest possible value of x? Triangle Side Lengths 5, 6, 15 5, 10, 15 5, 13, 15 5, 7, 15 5, 11, 15 5, 14, 15 5, 8, 15 5, 12, 15 5, 15, 15 5, 9, 15 The middle side can not be equal to 15. Answer is 14

In the right triangle ABC, segment DE is drawn from side to as shown, forming right triangle ADE. If is 24, is 12, and is 4, what is the length of ? 9 8 12 12  x = 24  8 x 12x = 192 4 x = 16 24

In the figure, the lengths of , , and are equal. x + w = 10 All sides equal Equilateral Triangle All angles equal Supplementary Angles (Sum of angles 180°) 60° x = 180 – 60 = 120 w = 180 – 60 = 120 x + w = 120+120 = 240 60° 60°

A C = 180° – 80° – 50° = 50° 80° AB = AC 2x – 12 = x – 3 –x –x In ABC, the measure of A is 80° and the measure of B is 50°. If the length of AB is 2x – 12 and the length of AC is x – 3, what is the length of AB? 11 A C = 180° – 80° – 50° = 50° 80° AB = AC 2x – 12 = x – 3 –x –x x – 12 = –3 50° 50° x = 9 B C AB = 2(9)–12 = 18 –12 = 6

12 In the figure, AB = BC = CA. What is the length of , if bisects ABC? Method #1 Use Pythagorean Theorem to find length of . a2 + b2 = c2 ?2 + 22 = 42 ? ?2 + 4 = 16 ?2 = 12 2

12 In the figure, AB = BC = CA. What is the length of , if bisects ABC? Method #2 60° Use 30° – 60° – 90° Right Triangle Rule 30° 60° x 2x 30° ? 60° 60° 2

x = 5x = 5(8) = 40 In the figure, what is the length of ? 2x 24 3x 45° 13 In the figure, what is the length of ? Use 45° – 45° – 90° Right Triangle Rule 45° x 2x 24 3x 45° Find length of = 5x 3x = 24 x = 8 = 5(8) = 40

14 In the isosceles right triangle ABC, leg equals 6. What is the length of ? 3x = 6 x = 2 6 = 5x = 5(2) = 10

15 In the figure, if ABC is an isosceles triangle, what is the length of ? Part 1 Find unknown sides of ∆ACD Use 30° – 60° – 90° Right Triangle Rule 30° 60° x 2x 60° 5 ADC = 180 - 90 - 30 ADC = 60

15 In the figure, if ABC is an isosceles triangle, what is the length of ? Part 2 Use ∆ABC to find length of . Note: ∆ABC is isosceles. ? 5

In the right ABC, the length of leg is 16 In the right ABC, the length of leg is and D is the midpoint of . Find the length of . x= = 3 30° C = 180° – A – B = 180° – 60° – 90° = 30°

17 In the figure, x = 60°, y = 60°, z = 30° and the length of is 2. What is the length of ? B = 90 A = 60 2 30° C = 180 – 90 – 60 C = 30 2 60° 30° 60° 60°

In the figure, ABC is a right isosceles triangle with In the figure, ABC is a right isosceles triangle with . If AD = 2, what is the length of ? 18 a2 + b2 = c2 x x2 + x2 = 22 2x2 = 4 x x2 = 2

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19 13 12 5

20 Find the tangent of K.

Find the tangent of K. a2 + b2 = c2 x2 + 242 = 512 x2 + 576 = 2601 20 Find the tangent of K. a2 + b2 = c2 x2 + 242 = 512 x2 + 576 = 2601 –576 –576 x2 = 2025 x = 45 x 45

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22 cos  = ?

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24 Find the length of JK. L 34.6 mm 18 K J 1  x = .9511  34.6 x x = 32.91 cos 18 = .9511

25 Find the length of FH.

tan 31 = .60 Find the length of FH. x 1  x = 10  0.60 x = 6.0 H G F 25 Find the length of FH. H x 31 G F 10 in. 1  x = 10  0.60 x = 6.0 tan 31 = .60