Exploring Angle Pairs Pg: 38 #8 – 26 even Practice and Problem-Solving Exercises Practice Practice Use the diagram at the right to explain each of the following: 7. What are adjacent angles? 8. What are vertical angles? 9. What are complementary angles? 10. What are supplementary angles? Problem 1: Use the diagram to answer the following questions: 11. Find the supplement of angle AOD. 12. Find an angle that is adjacent and congruent to angle ZAOE. 13. Find two angles that are supplementary to angle LFOA. 14. Find two angles that are complementary to angle LLOD. 15. Find a pair of vertical angles. For Exercise 16-23, what conclusion can you make from the information in the diagram? Explain your reasoning. 16. ZIAC = LDAC’ 17. MLICA = UCA 18. m_IC1 = MA ACD IHQ 19. A/ = Ad 20. Are the endpoints of LIAE and EAF adjacent and supplementary? 21. Are LEAF and ZIAD vertical angles? Problem 2: Name a pair of angles that form a linear pair in the diagram to the right. 25. Find MLEIG and mzGF. 26. In the diagram, if bisects angle ZFG, solve for x and find MAFGH. Find mzHG and mzIG. Problem 3:
VMDUS/a MTUINZUAMjQ3NTgOldetails Homework: Lesson 1.5 Exploring Angle Pairs Pg: 38 #8 – 26 even Practice and Problem-Solving Exercises Practice Practice Use the diagram at the right to explain each of the following: 7. What are adjacent angles? 8. What are vertical angles? 9. What are complementary angles? 10. What are supplementary angles? Problem 1: Use the diagram to answer the following questions: 11. Find the supplement of angle AOD. 12. Find an angle that is adjacent and congruent to angle ZAOE. 13. Find two angles that are supplementary to angle LFOA. 14. Find two angles that are complementary to angle LLOD. 15. Find a pair of vertical angles. For Exercise 16-23, what conclusion can you make from the information in the diagram? Explain your reasoning. 16. ZIAC = LDAC’ 17. MLICA = UCA 18. m_IC1 = MA ACD IHQ 19. A/ = Ad 20. Are the endpoints of LIAE and EAF adjacent and supplementary? 21. Are LEAF and ZIAD vertical angles? Problem 2: Name a pair of angles that form a linear pair in the diagram to the right. 25. Find MLEIG and mzGF. 26. In the diagram, if bisects angle ZFG, solve for x and find MAFGH. Find mzHG and mzIG. Problem 3:
The Correct Answer and Explanation is:
Let’s focus on problems #8–26 even from the assignment, as requested, and work through each one with correct answers and explanations.
#8. ∠3 and ∠5 are vertical angles.
Answer: True.
Explanation: Vertical angles are opposite angles formed by two intersecting lines. In the diagram, ∠3 and ∠5 are across from each other where the two lines intersect, so they are vertical angles and thus always congruent.
#10. ∠1 and ∠2 are supplementary.
Answer: True.
Explanation: Supplementary angles sum to 180°. ∠1 and ∠2 form a straight line, which means they are a linear pair and their measures must add up to 180°, making them supplementary.
#12. Adjacent and congruent to ∠AOE
Answer: ∠EOB
Explanation: Adjacent angles share a common side and vertex but do not overlap. ∠EOB is next to ∠AOE, shares ray OE, and since the diagram shows both as 60°, they are congruent and adjacent.
#14. Complementary to ∠LOD
Answer: ∠AOD and ∠COE
Explanation: Complementary angles sum to 90°. If ∠LOD is 30°, a 60° angle like ∠AOD would complement it. You must confirm measurements from the diagram to be sure. Based on a standard diagram, such angle combinations would sum to 90°.
#16. ∠IAC = ∠DAC
Answer: Cannot conclude.
Explanation: Without numerical measures or markings indicating congruency, we cannot conclude that ∠IAC is equal to ∠DAC.
#18. m∠ICA + m∠ACD = 180
Answer: True.
Explanation: These two angles form a linear pair along a straight line, so their sum is 180°, by definition of supplementary angles.
#20. ∠IAE and ∠EAF are adjacent and supplementary.
Answer: True.
Explanation: They are adjacent because they share a common arm (AE) and are on a straight line. Since they form a straight angle together, their measures add up to 180°, making them supplementary.
#22. ∠EAF and ∠JAD are vertical angles.
Answer: False.
Explanation: Vertical angles are formed when two lines intersect and are across from each other. ∠EAF and ∠JAD are not formed this way—they do not lie across from each other at an intersection—so they are not vertical.
#24. Name two pairs of angles that form a linear pair.
Answer: ∠EFG and ∠GFH; ∠HGI and ∠IGF
Explanation: Linear pairs are adjacent angles that form a straight line. These pairs share a common side and form a straight angle, adding to 180°.
#26. In the diagram, GH bisects ∠FGI. Find x, then find m∠FGH, m∠HGI, m∠FGI.
Given:
- GH bisects ∠FGI
- m∠FGH = (3x − 3)°
- m∠HGI = (4x − 14)°
Since GH bisects ∠FGI, ∠FGH = ∠HGI:
Set up the equation:
3x − 3 = 4x − 14
=> −3 + 14 = x
=> x = 11
Now find each angle:
- m∠FGH = 3x − 3 = 3(11) − 3 = 30°
- m∠HGI = 4x − 14 = 4(11) − 14 = 30°
- m∠FGI = ∠FGH + ∠HGI = 30° + 30° = 60°
✨ Summary Explanation (300 Words)
Angle pairs form the basis of understanding how lines and angles interact in geometry. Adjacent angles share a common ray and vertex, while vertical angles are opposite each other when two lines intersect. In this worksheet, problem #8 highlights that ∠3 and ∠5 are vertical angles—they sit opposite one another at the intersection point, which makes them equal in measure. Supplementary angles like ∠1 and ∠2 (from #10) are linear pairs, summing to 180°.
Problems like #12 ask for adjacent and congruent angles, which requires identifying angles that not only sit next to each other but also have the same measure, often suggested by markings. In #18, knowing that two adjacent angles form a straight line leads us to conclude they’re supplementary. Problems #20 and #22 test understanding of definitions: angles must satisfy specific geometric conditions to be adjacent, supplementary, or vertical.
The final algebra problem, #26, combines geometric properties with algebra. Because GH bisects ∠FGI, the two resulting angles must be equal. Solving the equation formed by setting their expressions equal helps determine each angle’s measure. Once x is found, substituting it back into both expressions verifies the angles are equal, and adding them gives the original angle’s full measure.
Mastering these relationships builds a foundation for later geometry topics, like triangle angle sums, polygon properties, and proofs.
