Angles made by a transversal: solve for x

Grade 10 Math Worksheet - Challenge Me

Circle the best answer.

2. 1.

   UR at G is 72°, UR at H is 72°, UL at J is (4x − 4)° and LL at K is (2x + 16)°. Which conclusion does Ari justify?

   • p ∥ q at t; then x = 28 makes UR at J 72°, matching UR at G, so t ∥ u
   • x = 28, but UL at J is 108° and UR at G is 72°, so t is not parallel to u
   • 4x − 4 = 2x + 16 gives x = 10, so u makes 144° with p: t is not parallel to u
   • t ∥ u already, since both cross the parallel lines p and q
3. 2.

   UR at G is 70°, LL at H is 72° and UR at J is 70°. Nothing is given about parallel lines. Which conclusion does Bo justify?

   Use the picture in question 1.

   • t is not parallel to u, since p is not parallel to q
   • Nothing about t and u can be shown unless p ∥ q
   • p ∥ q and t ∥ u, since 70° and 72° are close enough on a sketch
   • t ∥ u by corresponding angles along p; p is not parallel to q
4. 3.

   p ∥ q, and point X lies between them. G on p and K on q are both left of X. Esme finds m∠GXK. Which answer and reason are right?

   • 85°: a line ℓ through X parallel to p makes corresponding angles of 35° and 50°
   • 85°: a line ℓ through X parallel to p makes alternate interior angles of 35° and 50°
   • 95°: the angles of triangle GXK add to 180°
   • 50°: ∠GXK and the 50° at K are alternate interior angles
5. 4.

   t ∥ u is given. UR at G is (3x + 12)°, UR at J is (2x + 26)° and LR at H is (5x + 56)°. What does Jun conclude about p and q?

   • x = 14, so UR at G is 54° and UR at H is 180° − 126° = 54°: p ∥ q
   • 3x + 12 + 2x + 26 = 180 gives x = 28.4, so p and q cannot be parallel
   • x = 14, but LR at H is 126°, not 54°, so p is not parallel to q
   • p ∥ q follows at once from t ∥ u

6. Read this first

   Use this

   Each angle is named by its side of the crossed line and its side of t. Proof: 1. p ∥ r and q ∥ r (given). 2. Above p right of t = above r right of t, and above q right of t = above r right of t (corresponding angles). 3. Above p right of t = above q right of t (reason ?). 4. p ∥ q (reason ?).

7. 5.

   Kira proves two lines parallel to a third are parallel. Which reasons belong in steps 3 and 4?

   • Substitution; the converse of the same-side interior angles theorem
   • Substitution; the corresponding angles theorem
   • Vertical angles; the converse of the corresponding angles theorem
   • Substitution; the converse of the corresponding angles theorem

8. Use this

   "UR at H" means the angle above q and right of t at H; UL, LR and LL work the same way. Proof: 1. p ∥ q and t ⊥ p, so UR at G is 90° (given, definition of perpendicular). 2. UR at H = UR at G (reason ?). 3. UR at H = 90° (substitution). 4. t ⊥ q (definition of perpendicular).

9. 6.

   Luz proves that a line perpendicular to one of two parallel lines is perpendicular to the other. Which reason belongs in step 2?

   • Corresponding angles are equal, since p ∥ q
   • Vertical angles are equal
   • Any two right angles are equal
   • The converse of the corresponding angles theorem
10. 7.

    Milo hears: if an angle where t crosses p equals the matching angle where u crosses q, then p ∥ q. Which case is a counterexample?

    • UR at G = UR at J = 70°, with p ∥ q given as well
    • UR at G = LL at G = 70°
    • UR at G = UR at H = 70°
    • UR at G = UR at K = 70°, with UR at H = 80°
11. 8.

    Above p right of t: (2x + 10)°. Below p right of u: (3x + 5)°. Above p left of v: (3x + 5)°. Given t ∥ u, is v ∥ t, Pia asks?

    • No: 2x + 10 = 3x + 5 gives x = 5, so t makes 20° and v 160°
    • No: v's 104° is not t's 76°
    • Yes: x = 33, and v's 104° on the left means 76° on the right, like t
    • Not enough: the angles are on different sides of p, so they cannot be compared
12. 9.

    Ren proves the triangle angle sum with line v through X parallel to p, as in the sketch. Which reason gives the 50° at X?

    • Corresponding angles are equal, with t as the transversal
    • Vertical angles are equal at X, where t and v cross
    • The converse of the alternate interior angles theorem, since v ∥ p
    • Alternate interior angles are equal, with t as the transversal
13. 10.

    Above p: left of t 115°, right of t (3x + 5)°, left of u (x + 95)°. Below p right of v: (2x + 50)°. Which lines does Tala prove parallel?

    • t ∥ v: 3x + 5 = 2x + 50 gives x = 45, so both make 140°
    • None: no two of the angles 65°, 115° and 90° are equal
    • t ∥ u only: 65° and 115° are same-side interior angles
    • Not enough: x needs a second equation to be found