Alternate And Interior Angles

Solve for alternate and interior angles using the properties of transversals over parallel lines.

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Alternate Angles

Define alternate angles based on transversals.

Spotting Alternate Angles

When a transversal line crosses two other lines, it creates pairs of angles on opposite (alternate) sides of the transversal.

Look for the "Z" shape formed by the lines. The angles tucked inside the corners of that "Z" are alternate angles.

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Alternate and Interior Angles Visual

Fig 5.25 is essential to define the alternate interior angles.

$Alternate and Interior Angles$

Click to zoom

Alternate angles are on opposite sides of the transversal, while same-side interior angles share a side.

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Interior Angles

Define interior angles on the same side of a transversal.

Inside the Lines

When a transversal cuts across two parallel lines, the four angles located between the parallel lines are called interior angles.

Specifically, we often need to look at the interior angles on the same side of the transversal (for example, $\angle 3$ and $\angle 6$ ).

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Alternate & Interior Angle Rules

State the theorems for alternate and interior angles.

Rule 1: Alternate Angles

∠ a lt er na t e_{1} = ∠ a lt er na t e_{2}

Alternate angles formed by a transversal on parallel lines are exactly equal.

Rule 2: Interior Angles

∠ in t er i o r_{1} + ∠ in t er i o r_{2} = 18 0^{\circ}

Interior angles on the same side of the transversal always add up to 180° (they are supplementary).

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Example 1: Finding All 8 Angles

Reproduce Example 1 from page 15-16.

Problem

Parallel lines $l$ and $m$ are intersected by transversal $t$ . If $\angle 6$ is $135^\circ$ , what are the measures of the other seven angles?

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Example 2: Checking for Parallel Lines

Reproduce Example 2 from page 16.

Problem

Lines $l$ and $m$ are intersected by the transversal $t$ . If $\angle a$ is $120^\circ$ and $\angle f$ is $70^\circ$ , are lines $l$ and $m$ parallel to each other?

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Example 3: Your Turn

Faded example based on Example 3.

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In our problem, parallel lines $l$ and $m$ are intersected by a transversal $t$ , and we are given that $\angle 3$ is $50^\circ$ . We need to find the measure of $\angle 6$ . We know that $\angle 3$ and $\angle 6$ are angles located on the same side of the transversal. According to the rule for parallel lines, angles in this position must always add up to degrees. Therefore, we can set up the equation: $50^\circ + \angle 6 = 180^\circ$ . By subtracting 50 from 180, we find that the measure of $\angle 6$ is degrees.

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Interior Angles

Define interior angles on the same side of a transversal.

Inside the Lines

When a transversal cuts across two parallel lines, the four angles located between the parallel lines are called interior angles.

Specifically, we often need to look at the interior angles on the same side of the transversal (for example, $\angle 3$ and $\angle 6$ ).

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Example 1: Finding All 8 Angles

Reproduce Example 1 from page 15-16.

Problem

Parallel lines $l$ and $m$ are intersected by transversal $t$ . If $\angle 6$ is $135^\circ$ , what are the measures of the other seven angles?

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Example 2: Checking for Parallel Lines

Reproduce Example 2 from page 16.

Problem

Lines $l$ and $m$ are intersected by the transversal $t$ . If $\angle a$ is $120^\circ$ and $\angle f$ is $70^\circ$ , are lines $l$ and $m$ parallel to each other?

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Example 3: Your Turn

Faded example based on Example 3.

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