Angle Relationships and Proofs — Solutions

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1. Complementary and supplementary angles

   1. x and 37° complementary: x = 90 − 37 = 53° (complementary angles)
   2. x and 64° complementary: x = 90 − 64 = 26° (complementary angles)
   3. x and 112° supplementary: x = 180 − 112 = 68° (supplementary angles)
   4. x and 79° supplementary: x = 180 − 79 = 101° (supplementary angles)
   5. 3x + 42 = 180: 3x = 138, x = 46° (angles on a straight line)
   6. 55 + x + 48 = 180: x = 180 − 103 = 77° (angles on a straight line)

2. Vertically opposite angles

   1. 73° intersection: Vertically opposite = 73° (vertically opposite angles); adjacent angles = 180 − 73 = 107° each (angles on a straight line)
   2. (2x+15) = (3x−10): x = 25; each angle = 2(25)+15 = 65° (vertically opposite angles)
   3. (x+20) + (x+20) + 280 = 360: 2(x+20) = 80; x+20 = 40; x = 20 (angles at a point sum to 360°)

3. Parallel lines — angle relationships

   1. Corresponding to 58°: 58° (corresponding angles, PQ ∥ RS)
   2. Alternate to 83°: 83° (alternate angles, PQ ∥ RS)
   3. Co-interior with 115°: 180 − 115 = 65° (co-interior angles, PQ ∥ RS)
   4. 4x−6 = 2x+32: 2x = 38; x = 19 (corresponding angles, PQ ∥ RS)
   5. Co-interior (3x+20)+(2x+10)=180: 5x+30=180; x=30; angles = 110° and 70° (co-interior angles, PQ ∥ RS)
   6. Alternate 5x−8 = 3x+12: 2x=20; x=10; angle=42° (alternate angles, PQ ∥ RS)

4. Two-column geometric proofs

   1. Prove ∠EFD = 48°: ∠BEF = ∠EFD = 48° (alternate angles, AB ∥ CD)
   2. ∠MQP = 127° and ∠LQP: ∠MQP = ∠LQN = 127° (vertically opposite angles); ∠LQP = 180 − 127 = 53° (angles on a straight line)
   3. ∠APC with P between AB and CD: Draw EP ∥ AB ∥ CD. ∠PAE = 62° ⇒ ∠AEP = 62° (alternate, AB∥EP); ∠PCD = 54° ⇒ ∠EPC = 54° (alternate, EP∥CD); ∠APC = 62+54 = 116°
   4. AB ∥ CD ∥ EF ∥ GH transversal 71°: 71° at AB corresponds to 71° at GH (corresponding angles, AB∥GH); same-side = 180−71 = 109° (co-interior angles, AB∥GH)

5. Are the lines parallel?

   1. Co-interior 95° + 85° = 180°: Yes, parallel. Co-interior angles sum to 180°, which is the condition for parallel lines.
   2. Corresponding 73° and 74°: No, not parallel. Corresponding angles are not equal (73° ≠ 74°).
   3. Alternate both 61°: Yes, parallel. Alternate angles are equal, so the lines are parallel.
   4. Same-side 108° + 72° = 180°: Yes, parallel. Co-interior angles sum to 180°.

6. Multi-step angle proofs

   1. Three parallel lines, (3x+7) and (7x−3) co-interior at third: Angle at line 1 corresponds to angle at line 2 = (3x+7)°. Co-interior with angle at line 3: (3x+7)+(7x−3)=180; 10x+4=180; x=17.6; angle 1=59.8°, angle 3=120.2°
   2. ∠AEC = 95° proof: Draw EF through E with EF ∥ AB. ∠AEF = 55° (alternate angles, AB∥EF). ∠FEC = 40° (alternate angles, EF∥CD). ∠AEC = 55+40 = 95°.
   3. Two lines perpendicular to a third: Let lines l and m both be perpendicular to line n. Then the angle each makes with n is 90°. The corresponding angles formed are both 90° = equal, so l ∥ m (if corresponding angles are equal, lines are parallel).
   4. Converse of alternate angles: Both students are correct. The alternate angle theorem states that parallel lines produce equal alternate angles. Its converse — equal alternate angles imply parallel lines — is also a valid geometric theorem.

7. Algebraic angle problems using parallel lines

   1. AB∥CD, co-interior (5x+12)+(3x+28)=180: 8x+40=180; x=17.5; angle at AB = 99.5°, angle at CD = 80.5° (co-interior angles, AB∥CD)
   2. Adjacent angles at first line, corresponding at second: (4x−5)+(2x+35)=180 (angles on straight line); 6x+30=180; x=25. First angle = 95°. Corresponding at second = 6(25)−45=105°. Since 95°≠105° the lines are not parallel (the given expressions are inconsistent unless the problem allows non-parallel lines — the corresponding angles test fails).
   3. T between PQ∥RS; angle with PQ = 38°, angle with RS = 47°: Draw auxiliary line EF through T parallel to PQ and RS. Alternate angles: ∠with PQ side = 38° (alternate, PQ∥EF). Alternate angles: ∠with RS side = 47° (alternate, EF∥RS). ∠PTR = 38+47 = 85°

8. Angle relationships in geometric figures

   1. Three parallel lines, angle at l&sub1; = 65° (transversal 1), co-interior angle at l&sub3; = 110° (transversal 2): The co-interior angle at l&sub3; with respect to l&sub1; via transversal 2 is 110°; its supplementary pair = 180−110=70°. The angle between the transversals at their intersection is found by: angle at intersection = 180−65−70=45° (using angle sum in the triangle formed), or via exterior angle theorem. 45°
   2. Interior angle = 5 × exterior angle at same vertex: Interior + exterior = 180° (angles on a straight line). Let ext = e; 5e+e=180; e=30°; interior = 150°. A regular polygon with interior angles 150° has n = 360/(180−150) = 360/30 = 12 sides (regular 12-gon).
   3. Sum of angles on one side of transversal at two parallel-line intersections: At each intersection, the transversal forms 4 angles summing to 360°. The two angles on one side of the transversal at one intersection sum to 180° (angles on a straight line). Likewise at the second intersection. Total = 180+180 = 360°. Both students’ claims about the sum being 360 are correct by the straight-line angle property at each intersection point.

9. Formal proofs involving unknown angles

   1. AB∥CD; ∠AEF=(7k−4)°, ∠EFD=(5k+20)°: These are alternate angles (AB∥CD). Set equal: 7k−4=5k+20; 2k=24; k=12; each angle = 80°. Proof: Statement: ∠AEF=∠EFD (alternate angles, AB∥CD). Statement: 7k−4=5k+20 ⇒ k=12. Angle = 80°.
   2. Lines WX, YZ, PQ intersect at O; ∠WOP=40°, ∠POY=55°: ∠WOP=40° (given); ∠POY=55° (given); ∠YOX=∠WOP=40° (no — vertically opposite to... let’s name carefully). ∠WOP+∠POY+∠YOE=180 where E is the other half of WX. Angles on one side of WX: 40+55+∠YOX—wait: ∠WOY=40+55=95° (adjacent). ∠XOZ=95° (vertically opposite). ∠WOZ=180−95=85° (straight line). ∠XOY=85° (vertically opposite). Six angles: 40°, 55°, 85°, 40°, 55°, 85°; sum=360° ✓

10. Real-world angle reasoning

    1. Roof rafter at 28° to ridge (horizontal ceiling parallel to ridge): The rafter crosses two parallel horizontal lines (ridge and ceiling/wall plate level). The angle with the ridge is 28°. The angle with the wall plate = 180−28=152° (co-interior angles, ridge ∥ wall plate) — but the acute angle between the rafter and the wall plate = 28° (alternate angles, ridge ∥ wall plate). The rafter meets the wall at 28°.
    2. Parallel roads A and B; angles (2m+15)° and (4m−25)° corresponding: Corresponding angles equal (parallel roads): 2m+15=4m−25; 40=2m; m=20; angle at A = 55°; angle at B = 55° (corresponding, A∥B); co-interior at B = 180−55=125°
    3. Co-interior 62° and 118°, student 1 says parallel, student 2 says not: 62+118=180°. By the co-interior angle test: if co-interior angles sum to 180°, the lines ARE parallel. Student 1 is correct. The first student correctly applied the co-interior angle condition; the second student is wrong.