Question

Please assist, quickly! I am sorry for the bombardment!

5) Options for CF = BE, BF, BC, BD
7) Options for Congruence (4, 6, 5) ---> ASA, Hypotenuse-Leg, SAS, and AAS

Answer 1

5. BE

7. Hypotenuse leg

Explanation:

I had constructed it. See the attachment

Statement or construction (Reason in bracket)

1. AB ║ CD

2 Construct BE perpendicular to CD such that point E is on CD.

3. Construct CF perpendicular to AB such that point F is on AB.

4. m ∡CFB = m ∡BEC =90° (All °perpendicular measures 90°(2,3))

`\boxed{\tt Note:}` Since all angles are 90°, we can say that BECF is rectangle or square.

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. ΔBCF ≅ ΔCBE( Hypotenuse leg congruence (4,6,5))

`\boxed{\tt Note:}` The hypotenuse leg congruence (HL) theorem states that two right triangles are congruent if the hypotenuse and a leg of one triangle are congruent to the hypotenuse and a leg of the other triangle.

8. ∡FBC ≅ ∡ECB( Corresponding parts of congruent figures are congruent (7))