Question

lease Help!! And Fast!! Will give Brainliest to the Best Answers (75 points) --- The historical society hired an artist to restore a stained glass window panel. After studying the original drawings, the artist knows that panel ABCD is square. He also knows that FG is a perpendicular bisector of BC and BC ≅ BE. However, in order to restore the panel to match its original specifications, he needs to know the measure of ∠BED. Given: ABCD is a square FG ⊥ BC BC ≅ BE Step 1: Draw EC on the diagram. Use the given information to explain how you know △EGC ≅ △EGB. Add the appropriate notation to the diagram. (5 points) Step 2: Building on the information from Step 1, use the spaces below to prove that m∠BEC = 60°. Add the appropriate notation to the diagram. (5 points; 4 points for the proof, 1 point for the diagram) Statements Reasons 1. △EGC ≅ △EGB 1. Given 2. EB ≅ EC 3. Given 4. EB ≅ EC ≅ BC 5. m∠BEC = 60° Step 3: Next, use the spaces below to prove the measure of ∠ECD = 30°. Add the appropriate notation to the diagram. (5 points; 4 points for the proof, 1 point for the diagram) Given: ABCD is a square FG ⊥ BC BC ≅ BE △ECG ≅ △EBG △BEC is equilateral m∠BEC = 60° Prove: m∠ECD = 30° Statements Reasons 1. △BEC is equilateral 2. m∠GCE = 60° 3. ABCD is a square 4. Substitution Step 4: Find m∠BED. Show your work and explain your reasoning. Add the appropriate notation to the diagram. (6 points; 4 points for showing work and explaining reasoning; 1 point for final answer; 1 point for diagram) Given: ABCD is a square FG ⊥ BC BC ≅ BE △ECG ≅ △EBG △BEC is equilateral m∠BEC = 60° △ECD is isosceles Find: m∠BED

Answer 1

Answer:

$\angle BED = 135^\circ$

Step-by-step explanation:

First of all, let us do a construction in the given question image.

Let us connect the points E and C.

Please refer to the image attached in the answer area.

Step 1:

EC is drawn in the diagram.

Now, consider the triangles △EGC and △EGB.

1. Side BG = GC (G is the point on perpendicular bisector of AC)

2. $\angle EGB =\angle EGC$ (G is the point on perpendicular bisector of AC)

3. Line EG is common to both the triangles.

So,from SAS (Side Angle Side) congruence i.e. two sides and angle between them are equal in two triangles.

△EGC ≅ △EGB

Step 2:

Now, we know that △EGC ≅ △EGB, we can say that the corresponding sides CE and BE must be equal to each other i.e. CE $\cong$ BE .

And we are already given that BC $\cong$ BE.

$\therefore$ In $\triangle BEC$:

EB ≅ EC ≅ BC that means $\triangle BEC$ is an equilateral triangle.

Hence, every angle of $\triangle BEC$ must be equal to $60^\circ$.

$\therefore$ ∠BEC = 60°.

Step 3:

Let us find ∠ECD= ?

ABCD is a square, so $\angle BCD =90^\circ$

$\angle BCE + \angle ECD = 90^\circ\\\Rightarrow 60^\circ+ \angle ECD = 90^\circ\\\Rightarrow \angle ECD = 30^\circ$

Step 4:

We know that ABCD is a square, so side BC = CD.

We have proved in the previous step that BC = EC

$\therefore$ in $\triangle CED$, two sides are equal so it is isosceles triangle.

One angle ∠ECD= $30^\circ$

As it is an isosceles triangle, the other two angles will be equal.

Let them be equal to $x^\circ$.

The sum of all 3 angles of a triangle is equal to $180^\circ$

$x+x+30=180\\\Rightarrow x = 75^\circ$

OR

we can say $\angle CED =75^\circ$

Now, from the diagram, we can see the following:

$\angle BED = \angle BEC + \angle CED$

$\Rightarrow \angle BED = 60^\circ+75^\circ\\\Rightarrow \angle BED = 135^\circ$