Question

Give statements and reasons

Answer 1

A two column proof is presented as follows;

Statement
`{}` Reason

1. AB ║ DE, BD bisects AE
`{}` Given

2. ∠BAE = ∠AED, ∠ABD = ∠BDA
`{}` Alternate ∠s are equal

3.
`\overline {AC}` =
`\overline {EC}` ,
`\overline {BC}` =
`\overline {CD}`
`{}`
`{}` By definition of bisection of line AE by BD

4. ΔABC ≅ ΔEDC
`{}` By SAA, rule of congruency

Explanation:

Step 1. AB ║ DE, BD bisects AE
`{}` Given

Step 2. ∠BAE , ∠AED, and ∠ABD, ∠BDA
`{}` are pairs of alternate angles formed by the parallel lines, AB and DE and are therefore, equal

Step 3.
`\overline {AC}` =
`\overline {EC}` ,
`\overline {BC}` =
`\overline {CD}` The bisection of line gives two lines of equal length. The bisection of AE by BD gives,
`\overline {AC}` and
`\overline {EC}` where
`\overline {AC}` =
`\overline {EC}`

Similarly, the bisection of BD by AE gives,
`\overline {BC}` and
`\overline {CD}`, where
`\overline {BC}` =
`\overline {CD}`

Step 4. ΔABC ≅ ΔEDC
`{}` By the Side-Angle-Angle (SAA), rule of congruency, which states that two triangles having two angles and the corresponding non included sides of each triangle equal to the other, the two triangles are congruent.