Question

in the figure shown, (hcparallel gd) and (mangle abc = 100^{circ}). which of the following statements is false? a. (mangle cbe=80^{circ}) b. (mangle def = 80^{circ}) c. (angle deb) and (angle cbe) are corresponding angles. d. (angle cbe) and (angle geb) are alternate - interior angles.

Explanation:

Step1: Analyze linear - pair angles

Since $\angle ABC = 100^{\circ}$ and $\angle ABC$ and $\angle CBE$ form a linear - pair (sum of angles in a linear - pair is $180^{\circ}$), then $m\angle CBE=180 - 100=80^{\circ}$. So, statement a is true.

Step2: Use parallel - line properties

Because $HC\parallel GD$, corresponding angles are equal. $\angle CBE$ and $\angle DEF$ are corresponding angles. Since $m\angle CBE = 80^{\circ}$, then $m\angle DEF = 80^{\circ}$. So, statement b is true.

Step3: Identify corresponding angles

$\angle DEB$ and $\angle CBE$ are not corresponding angles. Corresponding angles are in the same relative position with respect to the parallel lines and the transversal. So, statement c is false.

Step4: Identify alternate - interior angles

$\angle CBE$ and $\angle GEB$ are alternate - interior angles (formed by two parallel lines $HC$ and $GD$ and a transversal $AB$). So, statement d is true.

Answer:

C. $\angle DEB$ and $\angle CBE$ are corresponding angles.