Question

tw bisects ∠uwy. complete the proof that △tvw ≅ △txw.
1 tw bisects ∠uwy
2 ∠v ≅ ∠x
3 ∠xwy ≅ ∠uvw
4 ∠twy ≅ ∠twu
5 m∠twx = m∠twy + m∠xwy
6 m∠twv = m∠twu + m∠uvw
7 m∠twx = m∠twu + m∠uvw
8 m∠twv = m∠twx
9 tw ≅ tw
10 △tvw ≅ △txw

Explanation:

Step1: Given

Statement 1 is given in the problem.

Step2: Given

Statement 2 is given in the problem.

Step3: Vertical angles are congruent

Vertical angles have equal measures.

Step4: Definition of angle bisector

Since \(\overleftrightarrow{TW}\) bisects \(\angle UWY\), by the definition of angle bisector, \(\angle TWY\cong\angle TWU\).

Step5: Angle addition postulate

For \(\angle TWX\), it is composed of \(\angle TWY\) and \(\angle XWY\), so \(m\angle TWX = m\angle TWY + m\angle XWY\) by the angle - addition postulate.

Step6: Angle addition postulate

For \(\angle TWV\), it is composed of \(\angle TWU\) and \(\angle UWV\), so \(m\angle TWV = m\angle TWU + m\angle UWV\) by the angle - addition postulate.

Step7: Substitution

Substitute \(m\angle TWY\) with \(m\angle TWU\) (from step 4) and \(m\angle XWY\) with \(m\angle UWV\) (from step 3) into \(m\angle TWX = m\angle TWY + m\angle XWY\).

Step8: Substitution

From step 7 (\(m\angle TWX=m\angle TWU + m\angle UWV\)) and step 6 (\(m\angle TWV = m\angle TWU + m\angle UWV\)), we get \(m\angle TWV=m\angle TWX\), so \(\angle TWV\cong\angle TWX\) (if two angles have equal measures, they are congruent).

Step9: Reflexive property of congruence

Any segment is congruent to itself, so \(\overline{TW}\cong\overline{TW}\).

Step10: AAS (Angle - Angle - Side) congruence criterion

In \(\triangle TVW\) and \(\triangle TXW\), we have \(\angle V\cong\angle X\) (step 2), \(\angle TWV\cong\angle TWX\) (step 8), and \(\overline{TW}\cong\overline{TW}\) (step 9). By the AAS (Angle - Angle - Side) congruence criterion, \(\triangle TVW\cong\triangle TXW\).

Answer:

1. Given
2. Given
3. Vertical angles are congruent
4. Definition of angle bisector
5. Angle addition postulate
6. Angle addition postulate
7. Substitution property of equality
8. Substitution property of equality (or transitive property of equality for angle measures)
9. Reflexive property of congruence
10. AAS (Angle - Angle - Side) congruence criterion