In the diagram of the rhombus ABCD, m C= 100 degrees. Use the CPCTC Theorem to prove that the diagonals BD and AC are perpendicular by finding the m D in DEC.

Question

GPT-4o mini · Answer

In a rhombus, all four sides are equal in length, and the diagonals bisect each other at right angles. Here, we will use the properties of a rhombus along with the given information about the angle $ m\angle C $ to find $ m\angle D $ and then prove that the diagonals $ AC $ and $ BD $ are perpendicular.

Properties of Angles in a Rhombus: In a rhombus, the opposite angles are equal, and the consecutive angles are supplementary (sum to $ 180^\circ $).

Given that $ m\angle C = 100^\circ $:

\[ m\angle A = m\angle C = 100^\circ \]

Since $ \angle A $ and $ \angle B $ are consecutive angles in rhombus $ ABCD $, we have:

\[ m\angle A + m\angle B = 180^\circ \]

Substituting the value of $ m\angle A $:

\[ 100^\circ + m\angle B = 180^\circ \]

Thus:

\[ m\angle B = 180^\circ - 100^\circ = 80^\circ \]

Since $ \angle B $ and $ \angle D $ are opposite angles:

\[ m\angle D = m\angle B = 80^\circ \]

Sum of Angles: Now, we have the four angles of rhombus $ ABCD $:

$ m\angle A = 100^\circ $
$ m\angle B = 80^\circ $
$ m\angle C = 100^\circ $
$ m\angle D = 80^\circ $

Using Diagonal Properties: The diagonals of a rhombus also bisect the angles at the vertices, which means:

\[ m\angle AOD = \frac{m\angle A}{2} = \frac{100^\circ}{2} = 50^\circ \] \[ m\angle BOC = \frac{m\angle B}{2} = \frac{80^\circ}{2} = 40^\circ \]

Using CPCTC: The angle formed at the intersection of the diagonals (let's denote it as $ \angle AOB $) is the sum of $ \angle AOD + \angle BOC $.

Thus,

\[ m\angle AOB = m\angle AOD + m\angle BOC = 50^\circ + 40^\circ = 90^\circ \]

Conclusion about Diagonals: Since $ m\angle AOB = 90^\circ $, we have proven that the diagonals $ AC $ and $ BD $ of rhombus $ ABCD $ are perpendicular by the property of angles and diagonal intersections.

Thus, by using the CPCTC theorem along with the properties of the rhombus, we conclude that $ AC \perp BD $.

Answer

It was actually 35 degrees

GPT-4o mini · Answer

Thank you for clarifying that $ m\angle C $ is $ 35^\circ $. Let's go through the steps again, $ m \angle C = 35^\circ $, to determine $ m \angle D $ and prove that the diagonals $ AC $ and $ BD $ are perpendicular. ### Step 1: Identify the Angles In rhombus $ ABCD $: - Opposite angles are equal. - Consecutive angles are supplementary. Since $ m\angle C = 35^\circ $: $$ m\angle A = m\angle C = 35^\circ $$ Now, using the fact that consecutive angles are supplementary: $$ m\angle A + m\angle B = 180^\circ $$ Substituting $ m\angle A $: $$ 35^\circ + m\angle B = 180^\circ $$ Thus, $$ m\angle B = 180^\circ - 35^\circ = 145^\circ $$ Since $ \angle B $ and $ \angle D $ are opposite angles: $$ m\angle D = m\angle B = 145^\circ $$ ### Step 2: Confirm All Angles Now we have the angles of rhombus $ ABCD $: - $ m\angle A = 35^\circ $ - $ m\angle B = 145^\circ $ - $ m\angle C = 35^\circ $ - $ m\angle D = 145^\circ $ ### Step 3: Use the Properties of Angles The diagonals of a rhombus bisect the angles at the vertices. Therefore, we can find the angles formed at the intersection of the diagonals: 1. **Angle at Vertex : $$ m\angle AOD = \frac{m\angle A}{2} = \frac{35^\circ}{2} = 17.5^\circ $$ 2. **Angle at Vertex B**: $$ m\angle BOC = \frac{m\angle B}{2} = \frac{145^\circ}{2} = 72.5^\circ $$ ### Step 4: Diagonal Angles At the intersection point (let’s name it $ O $), the angles formed by the diagonals are: $$ m\angle AOB = m\angle AOD + m\angle BOC = 17.5^\circ + 72.5^\circ = 90^\circ $$ ### Conclusion: Perpendicular Diagonals Since $ m\angle AOB = 90^\circ $, this demonstrates that the diagonals $ AC $ and $ BD $ are perpendicular. Thus, by using the properties of the rhombus, we have successfully proven that the diagonals of rhombus $ ABCD $ are perpendicular.