2. Lines, Angles, and Triangles
Geometry
Topic 2
Lines, Angles, and Triangles
Study

3. Reporting Category Questions

4. MAFS.912.G-CO.3.9
Using the figure below and the fact that line 𝑙 is parallel to segment 𝐴𝐶 prove that the sum of the angle measurements in a triangle is 180°.

5. Using the figure below, prove that vertical angles are congruent.
MAFS.912.G-CO.3.9
Using the figure below, prove that vertical angles are congruent.

6. Triangle 𝐴𝐵𝐶 and triangle 𝐿𝑀𝑁 are shown in the coordinate plane below.
MAFS.912.G-CO.2.8
Triangle 𝐴𝐵𝐶 and triangle 𝐿𝑀𝑁 are shown in the coordinate plane below.
Part A: Explain why triangle 𝐴𝐵𝐶 is congruent to triangle 𝐿𝑀𝑁 using one or more reflections, rotations, and translations.
Part B: Explain how you can use the transformations described in Part A to prove triangle 𝐴𝐵𝐶 is congruent to triangle 𝐿𝑀𝑁 by any of the criteria for triangle congruence (ASA, SAS, or SSS).

7. MAFS.912.G-CO.3.10
In the figure below, 𝐶𝐷 bisects ∠𝐴𝐶𝐵, 𝐴𝐵 = 𝐵𝐶 , 𝑚∠𝐵𝐸𝐶=90°, and 𝑚∠𝐷𝐶𝐸=42°. Find the measure of ∠𝐴.

8. MAFS.912.G-CO.3.10
In the figure below, 𝐴𝐷 is the angle bisector of ∠𝐵𝐴𝐶. 𝐵𝑃 and 𝐵𝐶 are straight lines, and 𝐴𝐷 ∥ 𝑃𝐶 . Prove that 𝐴𝑃=𝐴𝐶
Statements
Reasons

9. Describe a sequence of rigid transformations that shows △𝐴𝐵𝐶≅ △𝐷𝐸𝐹.
MAFS.912.G-CO.2.7 & MAFS.912.G-CO.2.8
△𝐴𝐵𝐶 and △𝐷𝐸𝐹, in the figure below are such that 𝐴𝐵 ≅ 𝐷𝐸 , 𝐴𝐶 ≅ 𝐷𝐹 , and ∠𝐴≅∠𝐷.
Which criteria for triangle congruence (ASA, SAS, SSS) implies that △𝐴𝐵𝐶≅ △𝐷𝐸𝐹?
Describe a sequence of rigid transformations that shows △𝐴𝐵𝐶≅ △𝐷𝐸𝐹.

10. MAFS.912.G-CO.3.10
Given that △𝐴𝐵𝐶≅ △𝑋𝑌𝑍, 𝐵𝐶=2(3𝑥−10), and 𝑌𝑍= −𝑥+15 ; find the value of 𝑥.

11. MAFS.912.G-CO.3.10
A billboard on level ground is supported by a brace, as shown in the accompanying diagram. The measure of angle A is 15° greater than twice the measure of angle B. Determine the measure of angle A and the measure of angle B.

12. Given △𝐾𝑃𝐿 is equilateral, 𝐽𝑃 ≅ 𝑀𝑃 , and ∠𝐽𝑃𝐾≅∠𝑀𝑃𝐿, prove △𝐽𝑃𝐾≅ △𝑀𝑃𝐿.
MAFS.912.G-CO.2.8
Given △𝐾𝑃𝐿 is equilateral, 𝐽𝑃 ≅ 𝑀𝑃 , and ∠𝐽𝑃𝐾≅∠𝑀𝑃𝐿,
prove △𝐽𝑃𝐾≅ △𝑀𝑃𝐿.
Statements
Reasons

13. MAFS.912.G-CO.3.9
Line n intersects lines l and m, forming the angles shown in the diagram below. Which value of x would prove 𝑙∥𝑚?
2.5
4.5
6.25
8.75

14. Given: C is the midpoint of 𝐵𝐸 . 𝐴𝐵 ≅ 𝐷𝐶 𝐴𝐵 ∥ 𝐷𝐶 , 𝐷𝐶 ⊥ 𝐵𝐸
MAFS.912.G-CO.2.8
Given:
C is the midpoint of 𝐵𝐸 .
𝐴𝐵 ≅ 𝐷𝐶
𝐴𝐵 ∥ 𝐷𝐶 , 𝐷𝐶 ⊥ 𝐵𝐸
Prove: ∠𝐴≅∠𝐷
Statements
Reasons

15. MAFS.912.G-CO.4.12
The picture to the right shows a construction of a line through a given point that is parallel to a given line. Which statement justifies why the constructed line is parallel to the given line?
When two lines are each perpendicular to a third line, the lines are parallel.
When two lines are each parallel to a third line, the lines are parallel.
When two lines are intersected by a transversal and alternate interior angles are congruent, the lines are parallel.
When two lines are intersected by a transversal and corresponding angles are congruent, the lines are parallel.

16. MAFS.912.G-CO.2.7
What additional information is required in order to prove the two triangles are congruent using the provided justification? Use the set of choices in the box below. Select a side or angle and place it in the appropriate region. Only one side or angle can be placed in each region. 𝐴𝐵 𝐴𝐶 𝐴𝐷 𝐵𝐶 𝐵𝐷 𝐶𝐷 𝐶𝐸 𝐷𝐸
∠𝐴𝐵𝐶
∠𝐴𝐵𝐷
∠𝐴𝐶𝐵
∠𝐴𝐷𝐵
∠𝐵𝐴𝐶
∠𝐶𝐷𝐸
∠𝐶𝐸𝐷
∠𝐷𝐶𝐸
ASA Postulate
SAS Theorem ≅ ≅

17. You wish to prove △𝑊𝑆𝑅≅ △𝑈𝑆𝑇
MAFS.912.G-SRT.2.5
Use the figure below, where point S is the midpoint of 𝑊𝑈 . Therefore 𝑊𝑆 ≅ 𝑈𝑆 .
You wish to prove △𝑊𝑆𝑅≅ △𝑈𝑆𝑇
In addition to 𝑊𝑆 ≅ 𝑈𝑆 , what two congruence statements are needed to prove this congruence by:
SSS
SAS
ASA
AAS
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18. MAFS.912.G-SRT.2.5
Consuela wants to determine the length of a power line that will be stretched over a lake. She cannot walk through the lake. She was able to take some measurements, hoping to determine the length of the power line. Her measurements are shown to the right.
Consuela believes that the length of the power line is 625 feet, but she’s not sure how to explain this to her boss.
Using what you know about triangle congruence, help Consuela by writing a paragraph proof to show why the length of the power line is 625 feet.
__________________________________________________________________________________________________________________________________________________

19. The lengths of two sides of a triangle are 2𝑛 and 𝑛−3 units, where
MAFS.912.G-CO.3.10
The lengths of two sides of a triangle are 2𝑛 and 𝑛−3 units, where
𝑛>3. Which inequality represents all possible lengths, 𝑥, for the third side of the triangle?
𝑛+3<𝑥<3𝑛−3
𝑛−3<𝑥<3𝑛+3
𝑛−3<𝑥<2𝑛
2𝑛<𝑥<3𝑛−3

20. Use this diagram to answer the following question.
MAFS.912.G-CO.3.10
Use this diagram to answer the following question.
What is the measure of ∠𝑄𝑃𝑅?
15°
60°
120°
175°

21. MAFS.912.G-CO.3.10
Given: 𝐴𝐵+𝐴𝐶>𝐵𝐶
Prove: 𝑥<7

22. Consider the two triangles shown.
MAFS.912.G-SRT.2.5
Consider the two triangles shown.
Part A
Write an inequality that shows the relationship between 𝑚∠𝐶 and 𝑚∠𝐹.
Part B
Write an inequality that shows the relationship between 𝑚∠𝐵+𝑚∠𝐴 and 𝑚∠𝐷+𝑚∠𝐸.

23. Connect point B and the arc with a straight edge.
MAFS.912.G-CO.4.12
Lewis is inscribing a square in a circle. He is at this stage, select the option that best describes his next step:
Connect point B and the arc with a straight edge.
Use the arcs and the center A to form a perpendicular bisector.
Using C as the center, draw an arc outside the circle.
Ignore the arcs drawn and draw another on the circumference of the circle. B

24. Two right triangles must be congruent if…
MAFS.912.G-CO.3.10
Two right triangles must be congruent if…
an acute angle in each triangle is congruent.
the lengths of the hypotenuses are equal.
the corresponding legs are congruent.
the areas are equal.

25. MAFS.912.G-GPE.2.5
Create the equation of the perpendicular bisector of 𝐴𝐵 , 𝐴(−4, 4) and 𝐵(4, 8) in slope intercept form.

26. MAFS.912.G-GPE.2.5
Determine the equation of the line that is parallel to 𝑦=−3𝑥+2 and goes through (1,5) in slope intercept form.

27. MAFS.912.G-GPE.2.5
Is triangle 𝑅𝑆𝑇, where 𝑅(4, 4), 𝑆(5, 1), 𝑇(−1, −1), a right triangle? If so, which angle is the right angle? Justify your answer.

28. MAFS.912.G-GPE.2.5
Line 𝐴 contains points (𝑝−4, 2) and (−2, 9). Line 𝐵 contains points (𝑝, −1) and (−1, 1). Find the value of 𝑝 if the lines are parallel.

29. MAFS.912.G-GPE.2.5
An equation of line 𝑎 is 𝑦=− 1 2 𝑥−2. Which is an equation of the line that is perpendicular to line 𝑎 and passes through the point (–4, 0)?
𝑦= − 1 2𝑥 +2
𝑦= − 1 2𝑥 +8
𝑦= 2𝑥−2
𝑦= 2𝑥+8

30. There is a line through 𝑍 that is parallel to 𝑋𝑌
MAFS.912.G-GPE.2.5
Use the three ordered pairs 𝑋 1, 0 , 𝑌 (10, 3), and 𝑍 (15, 4). Which of the following statements CANNOT be true?
There is a line through 𝑍 that is parallel to 𝑋𝑌
There is a line through 𝑍 that is perpendicular to 𝑋𝑌
There is a line through 𝑍 that is the same line as 𝑋𝑌
There is a line through 𝑍 that intersects 𝑋𝑌

31. Which of the following is NOT an equation for a line perpendicular to
MAFS.912.G-GPE.2.5
Which of the following is NOT an equation for a line perpendicular to
𝑦= 2 3 𝑥−1?
𝑦= −3/2𝑥−1
3𝑥+2𝑦=5
4𝑦=−6𝑥
2𝑦=−3/2𝑥+1