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Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Solve for the measure of ∠ACM in the parallelogram. Show your work and explain how you got your answer. **Diagram Explanation:** - The diagram provided is a parallelogram MONY. - Two diagonals, MY and ON, intersect at point C, forming four smaller triangles within the parallelogram. - The parallelogram has the following given angles and measures: - ∠NMY = 30° - ∠XNY = 40° - ∠MCY = 72° #### Solution Steps: 1. **Understanding Properties of Parallelograms:** Parallelograms have several properties that are useful for solving this problem: - Opposite sides are equal and parallel. - Opposite angles are equal. - Consecutive angles are supplementary. - Diagonals bisect each other, meaning they cut each other into two equal parts. 2. **Identify the Angles:** Given: - ∠NMY = 30° - ∠XNY = 40° - ∠MCY = 72° We need to find the measure of ∠ACM. 3. **Using the Properties of Triangles:** Since MY and ON intersect at C, they form several triangles within the parallelogram. Specifically, diagonal bisecting properties and internal triangle angle sum properties will be used. 4. **Finding ∠ACM:** The sum of the interior angles of any triangle is 180°. Given that in triangle MCY: - ∠MCY = 72° - ∠MYC = ∠ACM (same angle as ∠NMY due to opposite angles in parallel lines) This means: - ∠MCY + ∠MYC + ∠CMY = 180° - 72° + 30° + ∠CMY = 180° Solving for ∠CMY: - 102° + ∠CMY = 180° - ∠CMY = 180° - 102° = 78° Therefore, the measure of ∠ACM is 78°. By clearly understanding the properties of parallelograms and applying triangle angle sum properties, we can determine that the measure of ∠ACM is 78°.
Transcribed Image Text:### Solve for the measure of ∠ACM in the parallelogram. Show your work and explain how you got your answer. **Diagram Explanation:** - The diagram provided is a parallelogram MONY. - Two diagonals, MY and ON, intersect at point C, forming four smaller triangles within the parallelogram. - The parallelogram has the following given angles and measures: - ∠NMY = 30° - ∠XNY = 40° - ∠MCY = 72° #### Solution Steps: 1. **Understanding Properties of Parallelograms:** Parallelograms have several properties that are useful for solving this problem: - Opposite sides are equal and parallel. - Opposite angles are equal. - Consecutive angles are supplementary. - Diagonals bisect each other, meaning they cut each other into two equal parts. 2. **Identify the Angles:** Given: - ∠NMY = 30° - ∠XNY = 40° - ∠MCY = 72° We need to find the measure of ∠ACM. 3. **Using the Properties of Triangles:** Since MY and ON intersect at C, they form several triangles within the parallelogram. Specifically, diagonal bisecting properties and internal triangle angle sum properties will be used. 4. **Finding ∠ACM:** The sum of the interior angles of any triangle is 180°. Given that in triangle MCY: - ∠MCY = 72° - ∠MYC = ∠ACM (same angle as ∠NMY due to opposite angles in parallel lines) This means: - ∠MCY + ∠MYC + ∠CMY = 180° - 72° + 30° + ∠CMY = 180° Solving for ∠CMY: - 102° + ∠CMY = 180° - ∠CMY = 180° - 102° = 78° Therefore, the measure of ∠ACM is 78°. By clearly understanding the properties of parallelograms and applying triangle angle sum properties, we can determine that the measure of ∠ACM is 78°.
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