10 ab and cd are parallel. give a reason why a and d are equal. give a reason why b and e are equal. use your answers to a and b to show that the sum of the angles of triangle abc must be 180°.

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Accepted Answer

### Part 1: Reason why $ a $ and $ d $ are equal
# Brief Explanations:
$ AB \parallel CD $, and $ AC $ is a transversal. By the **alternate interior angles theorem**, alternate interior angles formed by a transversal cutting parallel lines are equal. Thus, $ a = d $.

# Answer:
$ a $ and $ d $ are equal because $ AB \parallel CD $ (given) and $ AC $ is a transversal; by the alternate interior angles theorem, alternate interior angles are equal.

### Part 2: Reason why $ b $ and $ e $ are equal
# Brief Explanations:
$ AB \parallel CD $, and $ BC $ is a transversal. By the **alternate interior angles theorem**, alternate interior angles formed by a transversal cutting parallel lines are equal. Thus, $ b = e $.

# Answer:
$ b $ and $ e $ are equal because $ AB \parallel CD $ (given) and $ BC $ is a transversal; by the alternate interior angles theorem, alternate interior angles are equal.

### Part 3: Show the sum of angles in $ \triangle ABC $ is $ 180^\circ $
# Explanation:
## Step 1: Recall angle relationships from parallel lines
From Parts 1 and 2, we know $ a = d $ and $ b = e $.

## Step 2: Analyze angles at point $ C $
At point $ C $, the angles $ d $, $ c $, and $ e $ form a straight line (linear pair), so their sum is $ 180^\circ $:
$$ d + c + e = 180^\circ $$

## Step 3: Substitute $ d $ with $ a $ and $ e $ with $ b $
Using $ d = a $ and $ e = b $, substitute into the equation:
$$ a + c + b = 180^\circ $$

## Step 4: Identify angles of $ \triangle ABC $
In $ \triangle ABC $, the angles are $ a $ (at $ A $), $ b $ (at $ B $), and $ c $ (at $ C $). Thus, the sum $ a + b + c = 180^\circ $, proving the angle - sum property of a triangle.

# Answer:
The sum of the angles of $ \triangle ABC $ is $ 180^\circ $ because $ d + c + e = 180^\circ $ (linear pair), and substituting $ d = a $ and $ e = b $ (from alternate interior angles) gives $ a + b + c = 180^\circ $, where $ a $, $ b $, and $ c $ are the angles of $ \triangle ABC $.

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Question

Explanation:

Part 1: Reason why \( a \) and \( d \) are equal

Step 1: Recall angle relationships from parallel lines

Step 2: Analyze angles at point \( C \)

Step 3: Substitute \( d \) with \( a \) and \( e \) with \( b \)

Step 4: Identify angles of \( \triangle ABC \)

Answer:

Part 2: Reason why \( b \) and \( e \) are equal