Question

angle relationships and parallel lines
example 7: determine the measure of each angle in the diagram below. label the
diagram.

m∠axb ______
m∠cxf ______
m∠axg ______
m∠gxe ______
m∠exo ______

example 8: determine the measure of each angle in the diagram below. label the
diagram.

m ① ______
m ② ______
m ③ ______

example 9: determine the measure of each angle in the diagram below. label the
diagram.

m∠1 ______
m∠2 ______
m∠3 ______
m∠4 ______
m∠5 ______
m∠6 ______

Explanation:

Example 7 Step 1: Analyze ∠AXB

∠AXB and the 32° angle with ∠CXG (right angle? Wait, ∠CXD is right? Wait, diagram has X with CG and EX perpendicular? Wait, ∠AXB: AXB is vertical to which? Wait, A and D are opposite rays, C and E are opposite rays (perpendicular? Because there's a right angle at X between C and D? Wait, the diagram: A---X---D (horizontal), C---X---E (vertical, right angle at X between C and D? So ∠CXD is 90°. Then ∠AXB: angle between A and B. The angle between B and C is 32°, so ∠AXB: since A and D are straight, C and E are straight (perpendicular), so ∠AXB + 32° + 90° + 40°? Wait no, A---X---D is straight (180°), C---X---E is straight (180°), and ∠CXD is 90° (right angle). So ∠AXB: let's see, the angle between BX and CX is 32°, CX and DX is 90°, DX and FX is 40°? Wait no, the diagram: B is above A, C is up, F is between C and D, D is right, A is left, G is below A, E is down. So ∠AXB: AXB is adjacent to 32° and the right angle? Wait, maybe ∠AXB: since A and D are straight, the angle at X: ∠AXB + 32° + 90° + 40° = 180°? No, A---X---D is 180°, so the angles on one side: ∠AXB (between A and B), ∠BXC (32°), ∠CXD (90°)? Wait no, ∠CXD is 90°? Wait the diagram has a right angle symbol at X between C and D? So ∠CXD = 90°. Then ∠BXC = 32°, ∠FXD = 40°, so ∠CXF = 90° - 40° = 50°? Wait, let's do Example 7 first.

Example 7 Step 1: m∠AXB

A---X---D is a straight line (180°). The angles at X: ∠AXB, ∠BXC (32°), ∠CXD (90°), and ∠DXF (40°)? Wait no, ∠CXD is 90° (right angle), so ∠BXC = 32°, so ∠AXB = 180° - 32° - 90° - 40°? Wait 32 + 90 + 40 = 162, 180 - 162 = 18? No, that can't be. Wait maybe A and G are opposite? Wait, G is below A, so A---X---D, G---X---F? Maybe vertical angles. Wait, ∠AXB: let's look at vertical angles. ∠AXB and ∠GXF? Wait, no, maybe ∠AXB: the angle between AX and BX. AX is left, BX is between AX and CX. The angle between BX and CX is 32°, and CX and EX are vertical (perpendicular to AX and DX). Wait, maybe I misread. Let's start over.

Diagram for Example 7:
- Horizontal line: A (left) --- X --- D (right)
- Vertical line: C (up) --- X --- E (down) (right angle at X between C and D, so ∠CXD = 90°)
- Ray B: between A and C, making 32° with C (∠BXC = 32°)
- Ray F: between C and D, making 40° with D (∠FXD = 40°)
- Ray G: between A and E, opposite to F?

So ∠AXB: angle between A (left) and B (between A and C). Since A---X---D is 180°, and ∠BXC = 32°, ∠CXD = 90°, so ∠AXB = 180° - 32° - 90° = 58°? Wait 32 + 90 = 122, 180 - 122 = 58. So m∠AXB = 58°.

Example 7 Step 2: m∠CXF

∠CXF is between C (up) and F (between C and D). ∠CXD = 90°, ∠FXD = 40°, so ∠CXF = 90° - 40° = 50°.

Example 7 Step 3: m∠AXG

∠AXG: A (left) and G (below A). Since G is opposite to F (vertical angles), ∠AXG should be equal to ∠CXF (50°)? Wait, or ∠AXG: A---X---D, G---X---F, so ∠AXG and ∠FXD? No, vertical angles: ∠AXG and ∠CXF? Wait, ∠AXG: angle between A and G. Since E is down, G is between A and E. The angle between A and B is 58°, B and C is 32°, C and E is 180° (vertical), so A and G: ∠AXG = ∠BXC = 32°? No, wait vertical angles: ∠AXG and ∠FXD? ∠FXD is 40°, so ∠AXG = 40°? Wait, maybe I messed up. Wait, A---X---D, G---X---F: so ∠AXG and ∠FXD are vertical angles? So ∠AXG = 40°.

Example 7 Step 4: m∠GXE

∠GXE: G (below A) and E (down). Since A---X---D is 180°, G---X---F is 180°, and E is down, C is up. ∠GXE: angle between G and E. Since E is vertical down, and G is below A, so ∠GXE: A---X---D (180°), so ∠AXG (40°) + ∠GXE + ∠EXD? No, E is down, so C---X---E is 180°, so ∠CXE = 180°, ∠BXC = 32°, ∠AXB = 58°, so ∠AXE: A---X---E? Wait, maybe ∠GXE: since G is below A, E is down, so ∠GXE is equal to ∠AXB (58°) because they are vertical angles? Wait, ∠AXB and ∠GXE: A---X---D, G---X---F, C---X---E. Maybe ∠GXE = 58°.

Example 7 Step 5: m∠EXD

∠EXD: E (down) and D (right). C---X---E is vertical (180°), ∠CXD = 90°, so ∠EXD = 90° (since C and E are vertical, D is right, so ∠EXD is 90°? Wait no, C---X---E is 180°, ∠CXD = 90°, so ∠EXD = 90° (because E is opposite C, so ∠EXD is vertical to ∠CXA? No, ∠EXD: E to X to D. Since C to X to D is 90°, E to X to D is also 90° (because C and E are opposite rays, so ∠EXD = ∠CXD = 90°).

Wait, maybe I made a mistake. Let's summarize Example 7:

- ∠AXB: 180° - 32° - 90° - 40°? No, 32 + 90 + 40 = 162, 180 - 162 = 18? No, that's wrong. Wait the right angle is between C and E? Wait the diagram has a right angle symbol at X between C and E? No, the right angle is between C and D. Wait, the right angle symbol is at X between C (up) and D (right), so ∠CXD = 90°. Then the angles on the top (C side): ∠BXC = 32°, ∠CXF =?, ∠FXD = 40°, so ∠CXF + 40° = 90° ⇒ ∠CXF = 50°. Then on the left (A side): ∠AXB + 32° + 90° = 180°? No, A---X---D is 180°, so ∠AXB + ∠BXC + ∠CXD = 180°? ∠AXB + 32° + 90° = 180° ⇒ ∠AXB = 58°. Then ∠AXG: since G is opposite F, ∠AXG = ∠FXD = 40° (vertical angles). ∠GXE: opposite to ∠AXB, so ∠GXE = 58° (vertical angles). ∠EXD: since E is opposite C, ∠EXD = ∠CXD = 90°? Wait no, E is down, D is right, so ∠EXD is 90°? Wait, C---X---E is 180°, so ∠CXD = 90° ⇒ ∠EXD = 90° (because E is straight down from C, so XE is vertical down, XD is horizontal right, so ∠EXD is 90°).

So Example 7:

m∠AXB = 58° (180 - 32 - 90 = 58)

m∠CXF = 50° (90 - 40 = 50)

m∠AXG = 40° (vertical to ∠FXD)

m∠GXE = 58° (vertical to ∠AXB)

m∠EXD = 90° (right angle, or vertical to ∠CXD)

Now Example 8: two parallel lines j and k, cut by a transversal. Angles ① (3x)°, ② (2x)°, ③ (y)°. Since j || k, consecutive interior angles? Wait, ① and ② are adjacent, forming a linear pair? Wait, ① and ② are on a straight line (transversal), so 3x + 2x = 180° (linear pair). So 5x = 180 ⇒ x = 36. Then ① = 3*36 = 108°, ② = 2*36 = 72°. Then ③ is equal to ① (corresponding angles) or ② (alternate interior)? Wait, j || k, so ③ = ① (corresponding angles) = 108°, or ③ = ② (alternate interior)? Wait, angle ② is (2x)° = 72°, angle ③: since it's a vertical angle or corresponding. Wait, the transversal cuts j and k, so angle ③ and angle ① are corresponding, so m③ = 108°, or angle ③ and angle ② are alternate interior, so m③ = 72°? Wait, no: angle ① and angle ② are on the transversal, linear pair (180°), so 3x + 2x = 180 ⇒ x=36. Then angle ① = 108°, angle ② = 72°. Then angle ③: since j || k, angle ③ is equal to angle ① (corresponding) or angle ② (alternate interior)? Wait, the diagram: angle ① is above j, angle ② is below j, same transversal. Then angle ③ is below k, same side as angle ②. So j || k, so angle ② and angle ③ are corresponding angles, so m③ = m② = 72°? Wait, no, corresponding angles: if j || k, then angle ② (on j, below transversal) and angle ③ (on k, below transversal) are corresponding, so equal. So m③ = 72°. Or angle ① (on j, above transversal) and angle ③ (on k, below transversal) are same-side exterior? No, better: linear pair: 3x + 2x = 180 ⇒ x=36. So ①=108°, ②=72°. Then ③=① (vertical angle? No, angle ③ is on k, so corresponding to ①: yes, because j || k, so corresponding angles are equal. So ③=108°. Wait, maybe I messed up. Let's see:

Example 8 Step 1: 3x + 2x = 180 (linear pair)

5x = 180 ⇒ x = 36

Step 2: m① = 3*36 = 108°

Step 3: m② = 2*36 = 72°

Step 4: m③ = m① (corresponding angles, since j || k) = 108° (or m③ = m②, alternate interior? Wait, no, angle ② is 72°, angle ③: if j || k, then angle ③ and angle ② are alternate interior angles, so equal. Wait, the diagram: angle ② is (2x)° on line j, below the transversal; angle ③ is (y)° on line k, below the transversal. So they are corresponding angles, so equal. So m③ = 72°? Wait, no, linear pair: ① and ② are 108 and 72, so ① is obtuse, ② is acute. Then angle ③: if it's on the other line, corresponding to ①, it's obtuse (108), corresponding to ②, acute (72). Let's check: j || k, transversal, so angle ② (72°) and angle ③ (y°) are alternate interior angles, so y = 72. Or angle ① (108°) and angle ③ (y°) are same-side exterior, supplementary? No, same-side exterior would be supplementary, but 108 + 72 = 180, so maybe angle ③ is equal to angle ① (108°) as corresponding angles. Wait, the diagram: angle ① is above j, angle ③ is below k? No, the diagram: angle ① is above the transversal, on line j; angle ② is below the transversal, on line j; angle ③ is below the transversal, on line k. So j || k, so angle ② (below j) and angle ③ (below k) are corresponding angles, so equal. So m③ = 72°.

Now Example 9: diagram with a right angle (vertical and horizontal lines), angles: (12y - 19)°, (6y + 1)°, (7x + 25)°, (8b - 7)°, (5b + 6)°, (8x - 10)°. Let's see, the right angle is between vertical (up) and horizontal (right), so the two angles (12y - 19)° and (6y + 1)° are complementary (sum to 90°). So 12y - 19 + 6y + 1 = 90 ⇒ 18y - 18 = 90 ⇒ 18y = 108 ⇒ y = 6. Then (12y - 19)° = 12*6 - 19 = 72 - 19 = 53°, (6y + 1)° = 36 + 1 = 37°. Then the horizontal line (left-right) has angles: (7x + 25)° and (8x - 10)° are vertical angles? Wait, (7x + 25)° and (8x - 10)°: since they are vertical angles, 7x + 25 = 8x - 10 ⇒ x = 35. Then (7x + 25)° = 7*35 + 25 = 245 + 25 = 270? No, that can't be. Wait, no, the horizontal line is left-right, vertical line is up-down, so the angles on the left (7x + 25)° and (8x - 10)°: maybe they are supplementary to the right angle? Wait, no, the vertical line (up-down) and horizontal line (left-right) form a right angle, so the angles on the left (7x + 25)° and (8x - 10)°: maybe they are vertical angles? Wait, (7x + 25)° and (8x - 10)°: 7x + 25 = 8x - 10 ⇒ x = 35, but 7*35 + 25 = 270, which is more than 180, impossible. So maybe (7x + 25)° and (8x - 10)° are adjacent, forming a linear pair (sum to 180°). So 7x + 25 + 8x - 10 =

Answer:

Example 7 Step 1: Analyze ∠AXB

∠AXB and the 32° angle with ∠CXG (right angle? Wait, ∠CXD is right? Wait, diagram has X with CG and EX perpendicular? Wait, ∠AXB: AXB is vertical to which? Wait, A and D are opposite rays, C and E are opposite rays (perpendicular? Because there's a right angle at X between C and D? Wait, the diagram: A---X---D (horizontal), C---X---E (vertical, right angle at X between C and D? So ∠CXD is 90°. Then ∠AXB: angle between A and B. The angle between B and C is 32°, so ∠AXB: since A and D are straight, C and E are straight (perpendicular), so ∠AXB + 32° + 90° + 40°? Wait no, A---X---D is straight (180°), C---X---E is straight (180°), and ∠CXD is 90° (right angle). So ∠AXB: let's see, the angle between BX and CX is 32°, CX and DX is 90°, DX and FX is 40°? Wait no, the diagram: B is above A, C is up, F is between C and D, D is right, A is left, G is below A, E is down. So ∠AXB: AXB is adjacent to 32° and the right angle? Wait, maybe ∠AXB: since A and D are straight, the angle at X: ∠AXB + 32° + 90° + 40° = 180°? No, A---X---D is 180°, so the angles on one side: ∠AXB (between A and B), ∠BXC (32°), ∠CXD (90°)? Wait no, ∠CXD is 90°? Wait the diagram has a right angle symbol at X between C and D? So ∠CXD = 90°. Then ∠BXC = 32°, ∠FXD = 40°, so ∠CXF = 90° - 40° = 50°? Wait, let's do Example 7 first.

Example 7 Step 1: m∠AXB

A---X---D is a straight line (180°). The angles at X: ∠AXB, ∠BXC (32°), ∠CXD (90°), and ∠DXF (40°)? Wait no, ∠CXD is 90° (right angle), so ∠BXC = 32°, so ∠AXB = 180° - 32° - 90° - 40°? Wait 32 + 90 + 40 = 162, 180 - 162 = 18? No, that can't be. Wait maybe A and G are opposite? Wait, G is below A, so A---X---D, G---X---F? Maybe vertical angles. Wait, ∠AXB: let's look at vertical angles. ∠AXB and ∠GXF? Wait, no, maybe ∠AXB: the angle between AX and BX. AX is left, BX is between AX and CX. The angle between BX and CX is 32°, and CX and EX are vertical (perpendicular to AX and DX). Wait, maybe I misread. Let's start over.

Diagram for Example 7:

• Horizontal line: A (left) --- X --- D (right)
• Vertical line: C (up) --- X --- E (down) (right angle at X between C and D, so ∠CXD = 90°)
• Ray B: between A and C, making 32° with C (∠BXC = 32°)
• Ray F: between C and D, making 40° with D (∠FXD = 40°)
• Ray G: between A and E, opposite to F?

So ∠AXB: angle between A (left) and B (between A and C). Since A---X---D is 180°, and ∠BXC = 32°, ∠CXD = 90°, so ∠AXB = 180° - 32° - 90° = 58°? Wait 32 + 90 = 122, 180 - 122 = 58. So m∠AXB = 58°.

Example 7 Step 2: m∠CXF

∠CXF is between C (up) and F (between C and D). ∠CXD = 90°, ∠FXD = 40°, so ∠CXF = 90° - 40° = 50°.

Example 7 Step 3: m∠AXG

∠AXG: A (left) and G (below A). Since G is opposite to F (vertical angles), ∠AXG should be equal to ∠CXF (50°)? Wait, or ∠AXG: A---X---D, G---X---F, so ∠AXG and ∠FXD? No, vertical angles: ∠AXG and ∠CXF? Wait, ∠AXG: angle between A and G. Since E is down, G is between A and E. The angle between A and B is 58°, B and C is 32°, C and E is 180° (vertical), so A and G: ∠AXG = ∠BXC = 32°? No, wait vertical angles: ∠AXG and ∠FXD? ∠FXD is 40°, so ∠AXG = 40°? Wait, maybe I messed up. Wait, A---X---D, G---X---F: so ∠AXG and ∠FXD are vertical angles? So ∠AXG = 40°.

Example 7 Step 4: m∠GXE

∠GXE: G (below A) and E (down). Since A---X---D is 180°, G---X---F is 180°, and E is down, C is up. ∠GXE: angle between G and E. Since E is vertical down, and G is below A, so ∠GXE: A---X---D (180°), so ∠AXG (40°) + ∠GXE + ∠EXD? No, E is down, so C---X---E is 180°, so ∠CXE = 180°, ∠BXC = 32°, ∠AXB = 58°, so ∠AXE: A---X---E? Wait, maybe ∠GXE: since G is below A, E is down, so ∠GXE is equal to ∠AXB (58°) because they are vertical angles? Wait, ∠AXB and ∠GXE: A---X---D, G---X---F, C---X---E. Maybe ∠GXE = 58°.

Example 7 Step 5: m∠EXD

∠EXD: E (down) and D (right). C---X---E is vertical (180°), ∠CXD = 90°, so ∠EXD = 90° (since C and E are vertical, D is right, so ∠EXD is 90°? Wait no, C---X---E is 180°, ∠CXD = 90°, so ∠EXD = 90° (because E is opposite C, so ∠EXD is vertical to ∠CXA? No, ∠EXD: E to X to D. Since C to X to D is 90°, E to X to D is also 90° (because C and E are opposite rays, so ∠EXD = ∠CXD = 90°).

Wait, maybe I made a mistake. Let's summarize Example 7:

• ∠AXB: 180° - 32° - 90° - 40°? No, 32 + 90 + 40 = 162, 180 - 162 = 18? No, that's wrong. Wait the right angle is between C and E? Wait the diagram has a right angle symbol at X between C and E? No, the right angle is between C and D. Wait, the right angle symbol is at X between C (up) and D (right), so ∠CXD = 90°. Then the angles on the top (C side): ∠BXC = 32°, ∠CXF =?, ∠FXD = 40°, so ∠CXF + 40° = 90° ⇒ ∠CXF = 50°. Then on the left (A side): ∠AXB + 32° + 90° = 180°? No, A---X---D is 180°, so ∠AXB + ∠BXC + ∠CXD = 180°? ∠AXB + 32° + 90° = 180° ⇒ ∠AXB = 58°. Then ∠AXG: since G is opposite F, ∠AXG = ∠FXD = 40° (vertical angles). ∠GXE: opposite to ∠AXB, so ∠GXE = 58° (vertical angles). ∠EXD: since E is opposite C, ∠EXD = ∠CXD = 90°? Wait no, E is down, D is right, so ∠EXD is 90°? Wait, C---X---E is 180°, so ∠CXD = 90° ⇒ ∠EXD = 90° (because E is straight down from C, so XE is vertical down, XD is horizontal right, so ∠EXD is 90°).

So Example 7:

m∠AXB = 58° (180 - 32 - 90 = 58)

m∠CXF = 50° (90 - 40 = 50)

m∠AXG = 40° (vertical to ∠FXD)

m∠GXE = 58° (vertical to ∠AXB)

m∠EXD = 90° (right angle, or vertical to ∠CXD)

Now Example 8: two parallel lines j and k, cut by a transversal. Angles ① (3x)°, ② (2x)°, ③ (y)°. Since j || k, consecutive interior angles? Wait, ① and ② are adjacent, forming a linear pair? Wait, ① and ② are on a straight line (transversal), so 3x + 2x = 180° (linear pair). So 5x = 180 ⇒ x = 36. Then ① = 336 = 108°, ② = 236 = 72°. Then ③ is equal to ① (corresponding angles) or ② (alternate interior)? Wait, j || k, so ③ = ① (corresponding angles) = 108°, or ③ = ② (alternate interior)? Wait, angle ② is (2x)° = 72°, angle ③: since it's a vertical angle or corresponding. Wait, the transversal cuts j and k, so angle ③ and angle ① are corresponding, so m③ = 108°, or angle ③ and angle ② are alternate interior, so m③ = 72°? Wait, no: angle ① and angle ② are on the transversal, linear pair (180°), so 3x + 2x = 180 ⇒ x=36. Then angle ① = 108°, angle ② = 72°. Then angle ③: since j || k, angle ③ is equal to angle ① (corresponding) or angle ② (alternate interior)? Wait, the diagram: angle ① is above j, angle ② is below j, same transversal. Then angle ③ is below k, same side as angle ②. So j || k, so angle ② and angle ③ are corresponding angles, so m③ = m② = 72°? Wait, no, corresponding angles: if j || k, then angle ② (on j, below transversal) and angle ③ (on k, below transversal) are corresponding, so equal. So m③ = 72°. Or angle ① (on j, above transversal) and angle ③ (on k, below transversal) are same-side exterior? No, better: linear pair: 3x + 2x = 180 ⇒ x=36. So ①=108°, ②=72°. Then ③=① (vertical angle? No, angle ③ is on k, so corresponding to ①: yes, because j || k, so corresponding angles are equal. So ③=108°. Wait, maybe I messed up. Let's see:

Example 8 Step 1: 3x + 2x = 180 (linear pair)

5x = 180 ⇒ x = 36

Step 2: m① = 3*36 = 108°

Step 3: m② = 2*36 = 72°

Step 4: m③ = m① (corresponding angles, since j || k) = 108° (or m③ = m②, alternate interior? Wait, no, angle ② is 72°, angle ③: if j || k, then angle ③ and angle ② are alternate interior angles, so equal. Wait, the diagram: angle ② is (2x)° on line j, below the transversal; angle ③ is (y)° on line k, below the transversal. So they are corresponding angles, so equal. So m③ = 72°? Wait, no, linear pair: ① and ② are 108 and 72, so ① is obtuse, ② is acute. Then angle ③: if it's on the other line, corresponding to ①, it's obtuse (108), corresponding to ②, acute (72). Let's check: j || k, transversal, so angle ② (72°) and angle ③ (y°) are alternate interior angles, so y = 72. Or angle ① (108°) and angle ③ (y°) are same-side exterior, supplementary? No, same-side exterior would be supplementary, but 108 + 72 = 180, so maybe angle ③ is equal to angle ① (108°) as corresponding angles. Wait, the diagram: angle ① is above j, angle ③ is below k? No, the diagram: angle ① is above the transversal, on line j; angle ② is below the transversal, on line j; angle ③ is below the transversal, on line k. So j || k, so angle ② (below j) and angle ③ (below k) are corresponding angles, so equal. So m③ = 72°.

Now Example 9: diagram with a right angle (vertical and horizontal lines), angles: (12y - 19)°, (6y + 1)°, (7x + 25)°, (8b - 7)°, (5b + 6)°, (8x - 10)°. Let's see, the right angle is between vertical (up) and horizontal (right), so the two angles (12y - 19)° and (6y + 1)° are complementary (sum to 90°). So 12y - 19 + 6y + 1 = 90 ⇒ 18y - 18 = 90 ⇒ 18y = 108 ⇒ y = 6. Then (12y - 19)° = 126 - 19 = 72 - 19 = 53°, (6y + 1)° = 36 + 1 = 37°. Then the horizontal line (left-right) has angles: (7x + 25)° and (8x - 10)° are vertical angles? Wait, (7x + 25)° and (8x - 10)°: since they are vertical angles, 7x + 25 = 8x - 10 ⇒ x = 35. Then (7x + 25)° = 735 + 25 = 245 + 25 = 270? No, that can't be. Wait, no, the horizontal line is left-right, vertical line is up-down, so the angles on the left (7x + 25)° and (8x - 10)°: maybe they are supplementary to the right angle? Wait, no, the vertical line (up-down) and horizontal line (left-right) form a right angle, so the angles on the left (7x + 25)° and (8x - 10)°: maybe they are vertical angles? Wait, (7x + 25)° and (8x - 10)°: 7x + 25 = 8x - 10 ⇒ x = 35, but 7*35 + 25 = 270, which is more than 180, impossible. So maybe (7x + 25)° and (8x - 10)° are adjacent, forming a linear pair (sum to 180°). So 7x + 25 + 8x - 10 =