Question

here is a diagram showing triangle abc and some transformations of triangle abc. on the left side of the diagram, triangle abc has been reflected across line ac to form quadrilateral abcd. on the right side of the diagram, triangle abc been rotated 180° using mid - point m as a center to form quadrilateral abce. using what you know about rigid transformations, side lengths and angle measures, label as many side lengths and angle measures as you can in quadrilaterals abcd and abce.

Explanation:

Step1: Recall properties of rigid - transformations

Rigid transformations (reflection and rotation) preserve side - lengths and angle - measures.

Step2: Analyze quadrilateral ABCD (reflection)

Since triangle ABC is reflected across line AC to form quadrilateral ABCD, $\triangle ABC\cong\triangle ADC$. So, $AB = AD = 2.7$, $BC=DC = 3.2$, $\angle B=\angle D = 64.3^{\circ}$, $\angle BAC=\angle DAC$, $\angle BCA=\angle DCA$.

Step3: Analyze quadrilateral ABCE (rotation)

Since triangle ABC is rotated 180° about mid - point M to form quadrilateral ABCE, $\triangle ABC\cong\triangle CEA$. So, $AB = CE = 2.7$, $BC = AE=3.2$, $\angle B=\angle E = 64.3^{\circ}$, $\angle BAC=\angle ECA$, $\angle BCA=\angle EAC$.

Answer:

In quadrilateral ABCD: $AB = AD = 2.7$, $BC = DC = 3.2$, $\angle B=\angle D = 64.3^{\circ}$.
In quadrilateral ABCE: $AB = CE = 2.7$, $BC = AE = 3.2$, $\angle B=\angle E = 64.3^{\circ}$.