Question

unit 4 lesson 3: two-column proofs (adapted from henrico)

1. given: \\(\overline{ab} \cong \overline{cb}\\)

\\(\overline{eb} \cong \overline{db}\\)
prove: \\(\triangle abe \cong \triangle cbd\\)

(there is a two - column proof table with \statements\ and
easons\ columns, and a hand - written entry \\(\overline{ab}\cong\overline{cb}\\) in the statements column and \given\ in the reasons column. also, there is a diagram of two triangles \\(abe\\) and \\(cbd\\) with a common vertex at b.)

1. given: \\(\angle xwy \cong \angle zwy\\)

\\(\angle xyw \cong \angle zyw\\)
prove: \\(\triangle wxy \cong \triangle wzy\\)

(there is a two - column proof table with \statements\ and
easons\ columns, and a diagram of a quadrilateral - like figure with vertices w, x, y, z, and two triangles \\(wxy\\) and \\(wzy\\) sharing the side wy.)

Explanation:

Problem 2:

Step1: List given congruent sides

We know \(\overline{AB} \cong \overline{CB}\) (given) and \(\overline{EB} \cong \overline{DB}\) (given). Also, \(\angle ABE\) and \(\angle CBD\) are vertical angles, so \(\angle ABE \cong \angle CBD\) (Vertical Angles Theorem).

Step2: Apply SAS congruence

In \(\triangle ABE\) and \(\triangle CBD\):

• \(\overline{AB} \cong \overline{CB}\) (given)
• \(\angle ABE \cong \angle CBD\) (vertical angles)
• \(\overline{EB} \cong \overline{DB}\) (given)

By the Side - Angle - Side (SAS) Congruence Postulate, \(\triangle ABE \cong \triangle CBD\).

Problem 3:

Step1: List given congruent angles

We are given that \(\angle XWY \cong \angle ZWY\) and \(\angle XYW \cong \angle ZYW\). Also, \(\overline{WY}\) is a common side to both \(\triangle WXY\) and \(\triangle WZY\), so \(\overline{WY} \cong \overline{WY}\) (Reflexive Property of Congruence).

Step2: Apply ASA congruence

In \(\triangle WXY\) and \(\triangle WZY\):

• \(\angle XWY \cong \angle ZWY\) (given)
• \(\overline{WY} \cong \overline{WY}\) (reflexive)
• \(\angle XYW \cong \angle ZYW\) (given)

By the Angle - Side - Angle (ASA) Congruence Postulate, \(\triangle WXY \cong \triangle WZY\).

For Problem 2, the two - column proof can be filled as follows:

Statements	Reasons
\(\overline{EB} \cong \overline{DB}\)	Given
\(\angle ABE \cong \angle CBD\)	Vertical Angles Theorem
\(\triangle ABE \cong \triangle CBD\)	SAS Congruence Postulate

For Problem 3, the two - column proof can be filled as follows:

Statements	Reasons
\(\angle XYW \cong \angle ZYW\)	Given
\(\overline{WY} \cong \overline{WY}\)	Reflexive Property of Congruence
\(\triangle WXY \cong \triangle WZY\)	ASA Congruence Postulate

Answer:

Step1: List given congruent angles

We are given that \(\angle XWY \cong \angle ZWY\) and \(\angle XYW \cong \angle ZYW\). Also, \(\overline{WY}\) is a common side to both \(\triangle WXY\) and \(\triangle WZY\), so \(\overline{WY} \cong \overline{WY}\) (Reflexive Property of Congruence).

Step2: Apply ASA congruence

In \(\triangle WXY\) and \(\triangle WZY\):

• \(\angle XWY \cong \angle ZWY\) (given)
• \(\overline{WY} \cong \overline{WY}\) (reflexive)
• \(\angle XYW \cong \angle ZYW\) (given)

By the Angle - Side - Angle (ASA) Congruence Postulate, \(\triangle WXY \cong \triangle WZY\).

For Problem 2, the two - column proof can be filled as follows:

Statements	Reasons
\(\overline{EB} \cong \overline{DB}\)	Given
\(\angle ABE \cong \angle CBD\)	Vertical Angles Theorem
\(\triangle ABE \cong \triangle CBD\)	SAS Congruence Postulate

For Problem 3, the two - column proof can be filled as follows:

Statements	Reasons
\(\angle XYW \cong \angle ZYW\)	Given
\(\overline{WY} \cong \overline{WY}\)	Reflexive Property of Congruence
\(\triangle WXY \cong \triangle WZY\)	ASA Congruence Postulate