Question

1. which statement does not demonstrate the corollary to the triangle exterior angle theorem?

options:

• ( mangle cde > mangle dae )
• ( mangle cde > mangle dea )
• ( mangle bed > mangle eda )
• ( mangle cde > mangle bed )

Explanation:

Step1: Recall the corollary to the triangle exterior angle theorem

The corollary states that an exterior angle of a triangle is greater than either of the non - adjacent interior angles.

Step2: Analyze each option

• Option 1: $\angle CDE$ is an exterior angle of $\triangle ADE$, and $\angle DAE$ is a non - adjacent interior angle. So by the corollary, $m\angle CDE>m\angle DAE$ holds.
• Option 2: $\angle CDE$ is an exterior angle of $\triangle ADE$, and $\angle DEA$ is a non - adjacent interior angle. So by the corollary, $m\angle CDE > m\angle DEA$ holds.
• Option 3: $\angle BED$ is an exterior angle of $\triangle ADE$, and $\angle EDA$ is a non - adjacent interior angle. So by the corollary, $m\angle BED>m\angle EDA$ holds.
• Option 4: $\angle CDE$ and $\angle BED$: $\angle CDE$ and $\angle BED$ are not related by the exterior - interior angle relationship of a triangle. $\angle CDE$ is an angle in quadrilateral $BCDE$ (or related to $\triangle ADE$), and $\angle BED$ is an angle in the same quadrilateral. There is no exterior - interior angle relationship between them as per the corollary of the triangle exterior angle theorem.

Answer:

$m\angle CDE > m\angle BED$ (the last option)