Final Answer:
1. Given
2. Alternate Interior Angles Theorem
3. Definition of Congruent Angles
4. Angle Addition Postulate / Straight Angle
5. Substitution Property
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Explanation:
Let’s walk through this step by step — like we’re filling in a puzzle.
Step 1: “△ABC with parallel lines AD and CB”
→ This is just telling us what we already know. So the reason is “Given”.
Step 2: “∠3 ≅ ∠4” and “∠2 ≅ ∠5”
→ These angles are on opposite sides of the slanted lines, inside the parallel lines. When you have parallel lines and a line crossing them (called a transversal), those inside angles that are across from each other are equal. That’s called the “Alternate Interior Angles Theorem”.
Step 3: “m∠3 = m∠4” and “m∠2 = m∠5”
→ If two angles are congruent (≅), that means their measures are equal (=). So we write “Definition of Congruent Angles”.
Step 4: “m∠4 + m∠1 + m∠5 = 180°”
→ Look at point A — angles 4, 1, and 5 sit right next to each other on a straight line. A straight line is always 180°. So this is because of the “Angle Addition Postulate” or “Straight Angle”.
Step 5: “m∠1 + m∠2 + m∠3 = 180°”
→ We already know angle 2 equals angle 5, and angle 3 equals angle 4. So we can swap them in! That’s called the “Substitution Property”. Now we’ve shown the three angles of the triangle add up to 180°.
That’s it! You proved the Triangle Sum Theorem using parallel lines.
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Teach the Topic:
This problem is about proving that all three angles inside any triangle add up to 180 degrees. We do this by drawing a line parallel to one side of the triangle and using what we know about angles and parallel lines.
Think of it like this:
Imagine you’re standing at one corner of a triangle. If you turn to face the next corner, then turn again to face the last corner, you’ve turned halfway around — which is 180 degrees. That’s why the angles add up to 180!
Simple Example:
Draw a triangle. Put a ruler above the top side so it’s parallel to the bottom side. Now look at the angles — you’ll see how the outside angles match the inside ones, and together they make a straight line. That’s your proof!
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Practice Task:
Given: △XYZ with parallel lines XW and YZ
Prove: m∠X + m∠Y + m∠Z = 180°
Fill in the reasons for each statement:
Statements | Reasons
1. △XYZ with parallel lines XW and YZ | 1. _____
2. ∠Y ≅ ∠WXY, ∠Z ≅ ∠WXZ | 2. _____
3. m∠Y = m∠WXY, m∠Z = m∠WXZ | 3. _____
4. m∠WXY + m∠X + m∠WXZ = 180° | 4. _____
5. m∠X + m∠Y + m∠Z = 180° | 5. _____
Parent Tip: Review the logic above to help your child master the concept of triangle angle sum worksheet.
