# Parallel Lines and the Angle in a Triangle

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Naming Angle Pairs in Parallel Lines.

• Using the sketch below, identify only one additional set of each angle pair. ​
• And also indicate if they are congruent or supplementary.

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To Establish the Sum of the Angles of a Triangle.

(A). Consider triangle ABC below and follow the instructions.

• Select the Midpoint tool and click on any two points or a segment to find the midpoint.
• Rotate triangle ABC 180° around the midpoint of side AC. ​
• Right click on the mid-point of AC and select Rename to label the new vertex D. ​
• Rotate triangle ABC 180° around the midpoint of side AB. ​
• Right click on the mid-point of AB and select Rename to label the new vertex E. ​
• Look at angles EABBAC, and CAD

(1). Without measuring, write what you think is the sum of the measures of these angles. Explain or show your reasoning.

(2). What is the sum of the measures of angles ABC, BAC, and ACB?

Use a protractor to measure the four angles inside the quadrilateral.

What is the sum of these four angle measures?

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To Prove that the Sum of the Angles of a Triangle is 180 degrees.

(A). Here is triangle △ABC. Line DE is parallel to line AC.

(1). What is m∠DBA + b + m∠CBE? Explain how you know.

(2). Use your answer to explain why a + b + c = 180.

(3). Explain why your argument will work for any triangle: that is, explain why the sum of the angle measures in any triangle is 180°.

(B). This diagram below shows a square BDFH that has been made by images of triangle ABC under rigid transformations.

Given that angle BAC measures 53 degrees, find as many other angle measures as you can.

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For each triangle, find the measure of the missing angle.

Is there a triangle with two right angles? Explain your reasoning.

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In the diagram below, lines AB and CD are parallel.

(1). What is mACE ?

(2). What is mACB ?

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Here below is a diagram of triangle DEF.

(1). Find the measures of angles q, r, and s.

(2). Find the sum of the measures of angles q, r, and s.

(3). What do you notice about these three angles?

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