Intermediate Mathematics

Lines and Angles

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Lines and Angles

Intersecting Lines - Intersecting lines are lines that meet at a point.
Vertical Angles - Vertical angles are formed when two lines intersect.
Adjacent Angles - Adjacent angles share a common ray and a common vertex.

Lines d and e are intersecting lines.
Angles 1 and 3 are vertical angles.
Angles 2 and 4 are vertical angles.
Angles 1 and 4 are adjacent angles.
Angles 3 and 4 are adjacent angles.

Lines and Angles

Congruent Angles - Congruent angles have the same measure.

The vertical angles formed by two intersecting lines are congruent.

Angles 1 and 3 are vertical angles. They are congruent. This can be written as ∠1 ≅ ∠3.
If ∠1 measures 120^° , then ∠3 measures 120^°.

Lines and Angles

The adjacent angles formed by two intersecting lines are supplementary.

Supplementary angles together form a straight angle, so they have a sum of 180^°.

Angle 1 and 2 are adjacent angles.
If ∠1 measures 120^° , then ∠2 measures 60^°.
(120^° + 60^° = 180^°)

Lines and Angles

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Which angles are vertical angles?

Question 1 of 10

Angles 1 and 3 are vertical angles. Vertical angles are congruent. The other set of vertical angles is angle 2 and angle 4.

Angles 2 and 3 are adjacent angles. Try again.

Angles 1 and 4 are adjacent angles. Try again.

Angles 3 and 4 are adjacent angles. Try again.

One pair of angles is vertical. Try again.

Vertical angles are formed when two lines intersect. They are congruent.

Lines and Angles

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Which angles are supplementary?

Question 2 of 10

Angles 2 and 4 are vertical angles. Vertical angles are congruent, not supplementary. Try again.

Angles 1 and 3 are vertical angles. Vertical angles are congruent, not supplementary. Try again.

Angles 2 and 3 are adjacent and supplementary. Together, they form a straight angle which measures 180^°.

Angles 1 and 3 are vertical angles. Vertical angles are congruent, not supplementary. Try again.

One pair of angles is supplementary. Try again.

Supplementary angles form a straight angle, so they have a sum of 180^°.

Lines and Angles

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If ∠3 measures 65^°, then ∠4 measures ______.

Question 3 of 10

Angles 3 and 4 are supplementary angles, so they have a sum of 180^°. 115^° + 65^° = 180^°

The sum of these supplementary angles is 180^°. Try again.

Angles 3 and 4 are not congruent angles. They are supplementary angles with a sum of 180^°. Try again.

Angle 3 and angle 4 are supplementary angles with a sum of 180^°. Try again.

One of these is the size of ∠4. Try again.

Angle 3 and angle 4 are supplementary angles. They have a sum of 180^°.

Lines and Angles

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If ∠1 measures 72^°, then ∠3 measures ______.

Question 4 of 10

Angles 1 and 3 are congruent, not supplementary angles. Try again.

Angles 1 and 3 are vertical. Vertical angles are congruent. So, if ∠1 measures 72^°, then ∠3 also measures 72^°.

Angles 1 and 3 are vertical angles. Vertical angles are congruent. Try again.

One of these is the size of ∠3. Try again.

Angle 1 and angle 3 are vertical angles. Vertical angles are congruent.

Lines and Angles

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If ∠2 measures 135^°, then ∠3 measures ________.

Question 5 of 10

Angles 2 and 3 are not congruent angles. They are supplementary angles with a sum of 180^°. Try again.

You may have made a mistake when subtracting. These two angles are supplementary. They have a sum of 180^°. Try again.

Angles 2 and 3 are adjacent, supplementary angles. Together they measure 180^°. Try again.

Together, angles 2 and 3 form a straight angle that measures 180^°.
135^° + 45^° = 180^°

One of these is the size of ∠3. Try again.

Angle 2 and angle 3 are supplementary angles. They have a sum of 180^°.

Lines and Angles

Parallel Lines - Two lines in the same plane that do not intersect are parallel.

Transversal - A line that crosses two or more other lines is a transversal.

Lines a and b are parallel lines.
Line c is a transversal.
Angles 1 and 3 are vertical angles.
Angles 5 and 7 are vertical angles.
Angles 1 and 2 are adjacent angles.
Angles 5 and 6 are adjacent angles.

Lines and Angles

Corresponding Angles - When a traversal intersects a pair of parallel lines, eight angles are formed.
Angles in the same relative position are corresponding angles.

Angles 1 and 5 are corresponding angles. They are above a parallel line and to the left of the transversal.

Angles 3 and 7 are corresponding angles. They are beneath a parallel line and to the right of the transversal.

Lines and Angles

Corresponding angles are congruent.

If ∠1 measures 70^°, then ∠5 measures 70^°.

Lines and Angles

Given the measure of any one of the eight angles formed by two parallel lines and a transversal, the measure of each of the other seven angles can be determined.

Angles 1 and 2 are adjacent, supplementary angles.
∠2 measures 110^°    ∠1 + ∠2 = 180^°
Angles 1 and 3 are vertical angles.
∠3 measures 70^°    ∠1 ≡ ∠3
Angles 1 and 4 are adjacent, supplementary angles.
∠4 measures 110^°    ∠1 + ∠4 = 180^°
Angles 1 and 5 are corresponding angles.
∠5 measures 70^°    ∠1 ≡ ∠5
Similarly, ∠6 = 110^°,∠7 = 70^°, and ∠8 = 110^°.

Lines and Angles

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Which angles are corresponding angles?

Question 6 of 10

Angle 5 is above the transversal, and angle 8 is below the transversal. Corresponding angles are in the same relative position. Try again.

Angle 2 is above the traversal and to the right of a parallel line. Angle 4 is below the transversal and to the left of a parallel line. Corresponding angles are in the same relative position. Try again.

Angle 4 and 8 are corresponding angles. They are in the same relative position-below the transversal and to the left of a parallel line.

Angle 1 is to the left of a parallel line, and angle 2 is to the right of a parallel line. Corresponding angles are in the same relative position. Try again.

One pair of angles is corresponding. Try again.

Corresponding angles are in the same relative position.

Lines and Angles

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Which angles are corresponding angles?

Question 7 of 10

Angles 7 and 8 are supplementary angles. Together they form a straight angle that measures 180^°. Try again.

Angle 1 is above the transversal and to the left of a parallel line. Angle 3 is below the transversal and to the right of a parallel line. Corresponding angles are in the same relative position. Try again.

Angles 2 and 6 are in the same relative position -above the transversal and to the right of a parallel line.

Angle 3 is below the transversal, and angle 5 is above the transversal. Corresponding angles are in the same relative position. Try again.

One pair of angles is corresponding. Try again.

Corresponding angles are in the same relative position.

Lines and Angles

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If ∠1 measures 125^°, then ∠5 measures ________.

Question 8 of 10

Angles 1 and 5 are not supplementary angles. They are in the same relative position. Try again.

Angles 1 and 5 are corresponding angles. Corresponding angles are congruent.

Angles 1 and 5 are corresponding angles. Corresponding angles are congruent. Try again.

Try again.Supplementary angles have a sum of 180^°. Angles 1 and 5 are corresponding angles. Try again.

One of these is the size of ∠5.

Corresponding angles are congruent.

Lines and Angles

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If ∠3 measures 97^°, then ∠8 measures _______.

Question 9 of 10

Angles 3 and 4 are supplementary, so ∠4 = 83^°. Angles 4 and 8 are corresponding, so ∠4 ≅ ∠8. Therefore, ∠8 = 83^°.

You may have made a mistake when subtracting.
180^° - 97^° = ? Try again.

Angles 3 and 8 are not corresponding. Angles 3 and 4 are supplementary. Angles 4 and 8 are corresponding, congruent angles. Try again.

Together, angles 3 and 4 form a straight angle. Angles 4 and 8 are corresponding, congruent angles. Try again.

One of these is the size of ∠8. Try again.

Angles 3 and 7 are corresponding, congruent angles. Angles 8 and 7 are supplementary.

Lines and Angles

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If ∠5 measures 78^°, then ∠2 measures ________.

Question 10 of 10

Angle 5 is adjacent to angle 6. Angles 2 and 6 are corresponding, congruent angles. Try again.

Angles 2 and 5 are not corresponding angles. Angles 5 and 6 are supplementary. Angles 2 and 6 are corresponding, congruent angles. Try again.

Angles 5 and 6 are supplementary, so ∠6 = 102^°. Angles 2 and 6 are corresponding, so ∠2 ≅ ∠6. Therefore, ∠2 = 102^°.

You may have made a mistake when subtracting. 180^° - 78^° = ? Try again.

One of these is the size of ∠2. Try again.

Angles 5 and 6 are supplementary. Angles 2 and 6 are corresponding, congruent angles.

Lines and Angles