Question

2 construct $overline{be}$ perpendicular to $overleftrightarrow{cd}$ such that point $e$ is on $overleftrightarrow{cd}$. 3 construct $overline{cf}$ perpendicular to $overleftrightarrow{ab}$ such that point $f$ is on $overleftrightarrow{ab}$. 4 $mangle cfb = mangle bec = 90^{circ}$ all perpendicular angles measure $90^{circ}$ (2, 3). 5 $cf = be$ any point on one parallel line is the same distance from the other line on a perpendicular transversal (1, 2, 3). 6 $bc = bc$ they are measures of the same segment. 7 $\triangle bcfcong\triangle cbe$ congruence (4, 6, 5) 8 $angle fbccongangle ecb$ corresponding parts of congruent figures are congruent (7).

Explanation:

Step1: Recall perpendicular - angle property

When two lines are perpendicular, the angle between them is $90^{\circ}$. So, since $\overline{BE}\perp\overleftrightarrow{CD}$ and $\overline{CF}\perp\overleftrightarrow{AB}$, $m\angle CFB = m\angle BEC=90^{\circ}$.

Step2: Recall distance - between - parallel - lines property

If two lines are parallel, the distance between them (measured along a perpendicular transversal) is constant. So, if the lines related to $AB$ and $CD$ are parallel, $CF = BE$.

Step3: Identify common side

$BC$ is a common side for $\triangle BCF$ and $\triangle CBE$. So, $BC = BC$.

Step4: Apply congruence criterion

We have two right - angled triangles $\triangle BCF$ and $\triangle CBE$ with $CF = BE$, $BC = BC$, and $\angle CFB=\angle BEC = 90^{\circ}$. By the Hypotenuse - Leg (HL) congruence criterion for right - angled triangles (in a right - angled triangle, if the hypotenuse and one leg are equal to the hypotenuse and one leg of another right - angled triangle, the two triangles are congruent), $\triangle BCF\cong\triangle CBE$.

Step5: Use corresponding - parts - of - congruent - triangles property

Since $\triangle BCF\cong\triangle CBE$, corresponding parts of congruent triangles are congruent. So, $\angle FBC\cong\angle ECB$.

Answer:

The steps show the logical progression from constructing perpendiculars to proving the congruence of two triangles and then showing the congruence of corresponding angles. The key concepts used are perpendicular - angle measures, distance between parallel lines, the reflexive property of equality for the common side, the HL congruence criterion for right - angled triangles, and the property of corresponding parts of congruent triangles.