Question 1

Three angles of a triangle are in the ratio 2:3:4. What is the measure of the largest angle?

Accepted Answer

Step 1: Let the three angles be 2x, 3x, and 4x. Step 2: Sum of angles in a triangle = 180°, so 2x + 3x + 4x = 180°. Step 3: 9x = 180°, therefore x = 20°. Step 4: The largest angle = 4x = 4 × 20° = 80°. Therefore, the answer is 80°.

Question 2

Two parallel lines are cut by a transversal. If one of the alternate interior angles is 72°, what is the other alternate interior angle?

Accepted Answer

Step 1: When a transversal cuts two parallel lines, alternate interior angles are equal. Step 2: If one alternate interior angle is 72°, the other alternate interior angle is also 72°. Therefore, the answer is 72°.

Question 3

Two lines intersect each other. If one of the angles formed is 65°, what is the measure of the angle adjacent to it on a straight line?

Accepted Answer

Step 1: When two lines intersect, adjacent angles on a straight line are supplementary (sum to 180°). Step 2: If one angle is 65°, the adjacent angle = 180° − 65° = 115°. Therefore, the answer is 115°.

Question 4

A line makes an angle of 35° with the horizontal. What is the angle it makes with the vertical?

Accepted Answer

Step 1: The angle between horizontal and vertical is always 90°. Step 2: If a line makes 35° with the horizontal, it makes 90° − 35° = 55° with the vertical. Therefore, the answer is 55°.

Question 5

An exterior angle of a triangle is 110°. If the two non-adjacent interior angles are in the ratio 3:2, what is the measure of the smaller non-adjacent interior angle?

Accepted Answer

Step 1: By the exterior angle theorem, an exterior angle equals the sum of the two non-adjacent interior angles. Step 2: So, the two non-adjacent interior angles sum to 110°. Step 3: Let them be 3x and 2x. Then 3x + 2x = 110°, so 5x = 110°, and x = 22°. Step 4: The smaller angle = 2x = 2 × 22° = 44°. Therefore, the answer is 44°.

Question 6

Two lines intersect to form four angles. If one angle is 48°, what is the sum of the two angles adjacent to it?

Accepted Answer

Step 1: When two lines intersect, they form four angles. Two pairs of vertically opposite angles are equal. Step 2: Adjacent angles are supplementary (sum to 180°). Step 3: If one angle is 48°, each adjacent angle = 180° − 48° = 132°. Step 4: Sum of the two adjacent angles = 132° + 132° = 264°. Therefore, the answer is 264°.

Question 7

Three lines meet at a point. If two of the angles formed are 85° and 120°, what is the measure of the third angle?

Accepted Answer

Step 1: When three lines meet at a point, the sum of all angles around that point is 360°. Step 2: Third angle = 360° − 85° − 120° = 360° − 205° = 155°. Therefore, the answer is 155°.

Question 8

Two intersecting lines form four angles. If one angle is 3 times another angle, what is the measure of the larger angle?

Accepted Answer

Step 1: When two lines intersect, adjacent angles are supplementary (sum to 180°). Step 2: Let the smaller angle = x. Then the larger angle = 3x. Step 3: x + 3x = 180°, so 4x = 180°, giving x = 45°. Step 4: Larger angle = 3 × 45° = 135°. Therefore, the answer is 135°.

Question 9

Two lines intersect such that one of the angles formed is 65°. What is the measure of the angle adjacent to it on a straight line?

Accepted Answer

Step 1: When two lines intersect, adjacent angles on a straight line are supplementary (sum to 180°). Step 2: If one angle is 65°, the adjacent angle = 180° − 65° = 115°. Therefore, the answer is 115°.

Question 10

In a triangle, one angle measures 48° and another measures 67°. What is the measure of the third angle?

Accepted Answer

Step 1: The sum of all angles in a triangle is 180°. Step 2: Third angle = 180° − 48° − 67° = 180° − 115° = 65°. Therefore, the answer is 65°.

Question 11

Two parallel lines are cut by a transversal. If one of the alternate interior angles is 72°, what is the measure of the other alternate interior angle?

Accepted Answer

Step 1: When two parallel lines are cut by a transversal, alternate interior angles are equal. Step 2: If one alternate interior angle is 72°, the other alternate interior angle is also 72°. Therefore, the answer is 72°.

Question 12

A line makes an angle of 35° with the horizontal. What angle does it make with the vertical?

Accepted Answer

Step 1: The horizontal and vertical directions are perpendicular to each other, forming a 90° angle. Step 2: If a line makes 35° with the horizontal, it makes 90° − 35° = 55° with the vertical. Therefore, the answer is 55°.

Question 13

Two lines intersect such that one of the angles formed is 3 times the adjacent angle. If a third line is drawn parallel to one of the original lines, what is the measure of the angle that the third line makes with the transversal cutting both original lines?

Accepted Answer

Step 1: When two lines intersect, adjacent angles are supplementary. Let one angle be x, then the adjacent angle is 3x. Step 2: x + 3x = 180°, so 4x = 180°, giving x = 45°. Step 3: The angles formed are 45° and 135°. Step 4: When a line parallel to one of the original lines is drawn and cut by a transversal, corresponding angles are equal. The transversal makes an angle of 45° with the parallel line (corresponding to the 45° angle at the intersection).

Question 14

A transversal cuts two parallel lines. The difference between an interior angle on one side and its corresponding co-interior angle on the same side is 36°. What is the measure of the smaller of these two angles?

Accepted Answer

Step 1: Co-interior angles (also called consecutive interior angles or allied angles) on the same side of a transversal cutting parallel lines are supplementary. Step 2: Let the two angles be A and B, where A + B = 180°. Step 3: Given that A - B = 36° (or B - A = 36°). Step 4: Solving: A + B = 180° and A - B = 36°. Adding: 2A = 216°, so A = 108°. Subtracting: 2B = 144°, so B = 72°. Step 5: The smaller angle is 72°.

Question 15

Two parallel lines are cut by two different transversals. The first transversal makes an angle of 65° with the first parallel line. The second transversal makes an angle of 50° with the second parallel line. What is the acute angle between the two transversals?

Accepted Answer

Step 1: The first transversal makes 65° with the first parallel line, so it makes 180° - 65° = 115° on the other side. Step 2: Since the lines are parallel, the first transversal makes 65° with the second parallel line as well (corresponding angles). Step 3: The second transversal makes 50° with the second parallel line. Step 4: The angle between the two transversals at their intersection can be found by considering the angles they make with the parallel line. Step 5: If one transversal makes 65° and the other makes 50° with the same parallel line, the angle between them is |65° - 50°| = 15° or 180° - 15° = 165°. Step 6: The acute angle is 15°.

Question 16

A transversal intersects two parallel lines. One of the alternate interior angles is (5x - 10)° and the other is (3x + 30)°. If a perpendicular is drawn from the point of intersection on the first parallel line to the second parallel line, what is the distance between the parallel lines if the perpendicular has length 12 cm?

Accepted Answer

Step 1: Alternate interior angles are equal when a transversal cuts parallel lines. Step 2: 5x - 10 = 3x + 30. Step 3: 2x = 40, so x = 20. Step 4: Each alternate interior angle = 5(20) - 10 = 90°. Step 5: Since the angles are 90°, the transversal is perpendicular to both parallel lines. Step 6: The perpendicular drawn from the intersection point on the first line to the second line has length 12 cm, which is the distance between the parallel lines. Step 7: Distance = 12 cm.

Question 17

Three lines meet at a point. The angles formed are in the ratio 2:3:4. A fourth line is drawn such that it bisects the largest angle. What is the measure of one of the angles formed by the bisector and one of the original lines forming the largest angle?

Accepted Answer

Step 1: Three lines meeting at a point form angles that sum to 360°. Step 2: Let angles be 2x, 3x, and 4x. Then 2x + 3x + 4x = 360°, so 9x = 360°, giving x = 40°. Step 3: The angles are 80°, 120°, and 160°. Step 4: The largest angle is 160°. Step 5: The bisector divides 160° into two equal parts of 80° each. Step 6: One of the angles formed by the bisector and an original line forming the largest angle is 80°.

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