Question 1

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 2

A transversal intersects two parallel lines. If one of the corresponding angles is 72°, what is the measure of the other corresponding angle?

Accepted Answer

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

Question 3

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 4

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 larger 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: Let the angles be 3x and 2x. Step 3: 3x + 2x = 110°. Step 4: 5x = 110°, therefore x = 22°. Step 5: The larger angle = 3x = 3 × 22° = 66°. Therefore, the answer is 66°.

Question 5

If two angles form a linear pair and one angle is 28° more than the other, what is the measure of the smaller angle?

Accepted Answer

Step 1: A linear pair of angles sum to 180°. Let the smaller angle be x. Step 2: The larger angle = x + 28°. Step 3: x + (x + 28°) = 180°. Step 4: 2x + 28° = 180°. Step 5: 2x = 152°, therefore x = 76°. Therefore, the answer is 76°.

Question 6

Two complementary angles are in the ratio 1:5. What is the measure of the smaller angle?

Accepted Answer

Step 1: Complementary angles sum to 90°. Let the angles be x and 5x. Step 2: x + 5x = 90°. Step 3: 6x = 90°, therefore x = 15°. Step 4: The smaller angle = 15°. Therefore, the answer is 15°.

Question 7

At a point, four angles are formed such that three of them are in the ratio 1:2:3. If the fourth angle is 60°, what is the measure of the largest of the first three angles?

Accepted Answer

Step 1: All angles around a point sum to 360°. Step 2: Three angles are in ratio 1:2:3, so let them be k, 2k, and 3k. The fourth angle is 60°. Step 3: k + 2k + 3k + 60° = 360° → 6k = 300° → k = 50°. Step 4: The largest of the first three angles = 3k = 3(50°) = 150°.

Question 8

A transversal intersects two parallel lines. If one of the alternate interior angles is 72°, what is the sum of the two co-interior angles on the same side of the transversal?

Accepted Answer

Step 1: When a transversal intersects two parallel lines, alternate interior angles are equal. So the other alternate interior angle is also 72°. Step 2: Co-interior angles (also called consecutive interior angles) are supplementary, meaning they sum to 180°. Step 3: Therefore, the sum of the two co-interior angles = 180°.

Question 9

In a triangle, the exterior angle at one vertex is 128°. If the two non-adjacent interior angles are in the ratio 3:5, 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. So the two non-adjacent interior angles sum to 128°. Step 2: Let the angles be 3x and 5x. Then 3x + 5x = 128°, so 8x = 128°, giving x = 16°. Step 3: The smaller angle = 3x = 3(16°) = 48°.

Question 10

Two lines intersect such that one of the angles formed is 65°. What is the measure of the obtuse angle adjacent to this angle?

Accepted Answer

Step 1: When two lines intersect, adjacent angles are supplementary (sum to 180°). Step 2: If one angle is 65°, the adjacent angle = 180° − 65° = 115°. Step 3: 115° is obtuse (greater than 90°), so the answer is 115°.

Question 11

A straight line is divided into three angles in the ratio 2:3:4. What is the measure of the largest angle?

Accepted Answer

Step 1: Angles on a straight line sum to 180°. Step 2: Let the angles be 2k, 3k, and 4k. Then 2k + 3k + 4k = 180°. Step 3: 9k = 180°, so k = 20°. Step 4: The largest angle = 4k = 4(20°) = 80°.

Question 12

A line makes an angle of 35° with the positive x-axis. Another line is perpendicular to this line. A third line makes an angle of 110° with the positive x-axis. What is the acute angle between the second and third lines?

Accepted Answer

Step 1: The first line makes 35° with the positive x-axis. Step 2: A line perpendicular to it makes an angle of 35° + 90° = 125° with the positive x-axis (or 35° - 90° = -55°, which is equivalent to 125°). Step 3: The third line makes 110° with the positive x-axis. Step 4: The angle between the second and third lines is |125° - 110°| = 15°. Since 15° < 90°, it is already acute.

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: Let the smaller angle be x. Then the adjacent angle is 3x. Since adjacent angles on a straight line sum to 180°, we have x + 3x = 180°, so 4x = 180°, giving x = 45°. Step 2: The two angles formed are 45° and 135°. Step 3: When a third line is drawn parallel to one of the original lines, by the property of corresponding angles with a transversal, the angle formed equals one of the original angles. The angle with the transversal is 45° (or its supplement 135°, but the acute angle is 45°).

Question 14

Three lines intersect at a single point, forming six angles around that point. If the angles are in the ratio 2:3:4:2:3:4 (in order around the point), what is the measure of the largest angle?

Accepted Answer

Step 1: Angles around a point sum to 360°. Step 2: The angles are in the ratio 2:3:4:2:3:4. The sum of ratio parts is 2 + 3 + 4 + 2 + 3 + 4 = 18. Step 3: Each part represents 360° ÷ 18 = 20°. Step 4: The largest angle corresponds to the ratio part 4, so the largest angle is 4 × 20° = 80°.

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 a 65° angle with the first parallel line. Since the lines are parallel, it also makes a 65° angle with the second parallel line (corresponding angles). Step 2: The second transversal makes a 50° angle with the second parallel line. Step 3: The angle between the two transversals can be found by considering their angles with respect to the parallel lines. If the first transversal makes 65° and the second makes 50° with the same parallel line, the angle between them is |65° - 50°| = 15° or 180° - 15° = 165°. The acute angle is 15°.

Question 16

In a geometric figure, two lines intersect forming four angles. The bisectors of two adjacent angles are drawn. What is the angle between these two bisectors?

Accepted Answer

Step 1: When two lines intersect, they form four angles. Let's call two adjacent angles α and β. Since they are adjacent angles on a straight line, α + β = 180°. Step 2: The bisector of angle α divides it into two angles of α/2 each. The bisector of angle β divides it into two angles of β/2 each. Step 3: The angle between the two bisectors is the angle formed at their intersection point. Consider the triangle formed by the two bisectors and the point of intersection of the original lines. The angle between the bisectors is (α/2) + (β/2) = (α + β)/2 = 180°/2 = 90°.

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