Question

the circle centered at point a has a radius of length ab. the circles centered at points b and d have radii of length de. complete the proof that ∠dae ≅ ∠bae. step statement reason 1 ab = ad all radii of the same circle have the same length. 2 be = de both circles have radii of the same length. 3 ae = ae theyre lengths of the same segment. 4 side - side - side congruence (1, 2, 3) 5 corresponding parts of congruent triangles are congruent (4). options for step 4 and 5 as shown in the image.

Explanation:

Step 1: Analyze Step 4

We know from Step 1: \(AB = AD\), Step 2: \(BE = DE\), and Step 3: \(AE=AE\). By the Side - Side - Side (SSS) congruence criterion, if three sides of one triangle are equal to three sides of another triangle, the triangles are congruent. So the two triangles here are \(\triangle ABE\) and \(\triangle ADE\) because \(AB = AD\), \(BE=DE\), and \(AE = AE\). So the statement for Step 4 should be \(\triangle ABE\cong\triangle ADE\).

Step 2: Analyze Step 5

Once we have \(\triangle ABE\cong\triangle ADE\) (from Step 4), by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem, the corresponding angles \(\angle DAE\) and \(\angle BAE\) are congruent. The corresponding angles are the angles opposite the corresponding sides. In \(\triangle ABE\) and \(\triangle ADE\), \(\angle BAE\) and \(\angle DAE\) are the angles between \(AB\) & \(AE\) and \(AD\) & \(AE\) respectively, so they are corresponding angles.

Answer:

Step 4: \(\boldsymbol{\triangle ABE\cong\triangle ADE}\)

Step 5: \(\boldsymbol{\angle DAE\cong\angle BAE}\) (by CPCTC from \(\triangle ABE\cong\triangle ADE\))