Proofs About Angles Practice

Question

Complete this assessment to review what you’ve learned. It will not count toward your grade.
4 of 54 of 5 Items

Question
Use the image to answer the question.

An image is shown of perpendicular bisector VZ intersecting line segment WY at point X. Point U is a point on the perpendicular bisector.

Harriet has written a proof to prove that point U is equidistant from points W and Y. She knows that ZV¯¯¯¯¯¯¯¯
is the perpendicular bisector of WY¯¯¯¯¯¯¯¯¯
.

Statement Reason
1. ZV¯¯¯¯¯¯¯¯
is the perpendicular bisector of WY¯¯¯¯¯¯¯¯¯
. given
2. WX¯¯¯¯¯¯¯¯¯¯≅YX¯¯¯¯¯¯¯¯
definition of a perpendicular bisector
3. ∠WXU
and ∠YXU
are right angles. definition of a perpendicular bisector
4. ∠WXU≅∠YXU
Right Angle Congruence Theorem
5. XU¯¯¯¯¯¯¯¯≅XU¯¯¯¯¯¯¯¯
Reflexive Property of Congruence
6. UW¯¯¯¯¯¯¯¯¯≅UY¯¯¯¯¯¯¯¯
CPCTC Theorem

What is the missing step in her proof?

Option 1: There is no missing step in Harriet's proof.

Option 2: Harriet forgot the step where she proves that ∠XUW≅∠XUY
because of the CPCTC Theorem.

Option 3: Harriet forgot the step where she proves that △WXU≅△YXU
because of the SAS Congruence Theorem.

(1 point)
The missing statement in Harriet's proof is Option
.

Answer 1

To determine the missing step in Harriet's proof, let's analyze the proof steps provided:

1. The first statement asserts that \( ZV \) is the perpendicular bisector of \( WY \), which is given.
2. The second statement establishes that \( WX \cong YX \) based on the definition of a perpendicular bisector.
3. The third statement mentions that \( \angle WXU \) and \( \angle YXU \) are right angles as per the definition of a perpendicular bisector.
4. The fourth statement concludes that \( \angle WXU \cong \angle YXU \) due to the Right Angle Congruence Theorem.
5. The fifth statement is that \( XU \cong XU \) based on the Reflexive Property of Congruence.
6. The final statement establishes that \( UW \cong UY \) using the CPCTC Theorem, which is based on the congruence of triangles.

Now, to analyze the options for the missing step:

• Option 1: "There is no missing step in Harriet's proof." This is unlikely since proofs typically require a transition from angle congruence to triangle congruence.

• Option 2: "Harriet forgot the step where she proves that \( \angle XUW \cong \angle XUY \) because of the CPCTC Theorem." This statement does not quite fit since \( \angle XUW \) and \( \angle XUY \) are not directly relevant to establishing the triangle congruency needed.

• Option 3: "Harriet forgot the step where she proves that \( \triangle WXU \cong \triangle YXU \) because of the SAS Congruence Theorem." This option is correct because to use CPCTC, Harriet needs to establish that triangles \( WXU \) and \( YXU \) are congruent. She has the necessary components (two sides and the included angle) to show this:

  • \( WX \cong YX \) (from step 2)
  • \( XU \cong XU \) (the reflexive property from step 5)
  • \( \angle WXU \cong \angle YXU \) (from step 4)

Thus, using the Side-Angle-Side (SAS) Congruence Theorem, she could prove that \( \triangle WXU \cong \triangle YXU \).

Therefore, the most appropriate answer is:

Option 3: Harriet forgot the step where she proves that \( \triangle WXU \cong \triangle YXU \) because of the SAS Congruence Theorem.