For the three-part question that follows, provide your answer to each part in the given workspace. Identify each part with a coordinating response. Part A: State the corollary to t... For the three-part question that follows, provide your answer to each part in the given workspace. Identify each part with a coordinating response. Part A: State the corollary to the triangle exterior angle theorem. Part B: Identify the largest angle in triangle DEA. Explain. Part C: Place angles DEA, EAD, and ADE in order from least to greatest. Explain.
Understand the Problem
The question is asking for three parts related to the triangle and its properties. Part A requests the statement of the corollary to the triangle exterior angle theorem. Part B asks to identify the largest angle in triangle DEA and explain why. Part C requires placing the angles DEA, EAD, and ADE in order from least to greatest and to explain that as well.
Answer
Part A: The corollary states that the measure of an exterior angle is greater than either of the non-adjacent interior angles. Part B: The largest angle in triangle $DEA$ is $\angle EAD$. Part C: The order from least to greatest is $\angle DEA < \angle ADE < \angle EAD$.
Answer for screen readers
- Part A: The corollary states that the measure of an exterior angle is greater than either of the non-adjacent interior angles.
- Part B: The largest angle in triangle (DEA) is ( \angle EAD ).
- Part C: The order from least to greatest is ( \angle DEA < \angle ADE < \angle EAD ).
Steps to Solve
- Statement of the Corollary to the Triangle Exterior Angle Theorem
The corollary to the triangle exterior angle theorem states that the measure of an exterior angle of a triangle is greater than the measure of either of the non-adjacent interior angles. This can be expressed mathematically for triangle (ABC) with exterior angle (D) as:
$$ \angle D > \angle A \quad \text{and} \quad \angle D > \angle B $$
- Identifying the Largest Angle in Triangle DEA
Given the sides of triangle (DEA) are related as (DA > AE > ED), we can conclude that the largest angle is opposite the longest side. Here, (DA) is the longest side, hence:
$$ \angle EAD \text{ is the largest angle.} $$
This is due to the property that in any triangle, the side lengths are proportional to the opposite angles.
- Placing Angles in Order from Least to Greatest
To order the angles in triangle (DEA) from least to greatest based on the sides:
- Since (ED) is the shortest side, the angle opposite (ED) (which is ( \angle DEA )) is the smallest.
- The angle opposite (AE) (which is ( \angle ADE )) is next, and since (DA) is the longest side, opposite to (DA) (which is ( \angle EAD )) is the largest.
Thus the order from least to greatest is:
$$ \angle DEA < \angle ADE < \angle EAD $$
- Part A: The corollary states that the measure of an exterior angle is greater than either of the non-adjacent interior angles.
- Part B: The largest angle in triangle (DEA) is ( \angle EAD ).
- Part C: The order from least to greatest is ( \angle DEA < \angle ADE < \angle EAD ).
More Information
The triangle's properties reveal that the relationship between side lengths and angles is fundamental in understanding geometry. The corollary emphasizes how exterior angles relate to interior angles, while the sides determine the largest angle in a triangle.
Tips
- Confusing the sides with the angles: Remember that the longest side always corresponds to the largest angle.
- Misapplying the triangle inequality: Ensure that the relationships between sides are properly understood.
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