Question

If DG is the bisector of the angle D in the ADEF and DE = 6, DF - 8. EF 10, then
EG = ?​

Answer 1

Answer:

We say that triangle ABC is congruent to triangle DEF if

AB = DE

BC = EF

CA = FD

Angle A = Angle D

Angle B = Angle E

Angle C = Angle F

(Of course Angle A is short for angle BAC, etc.)

Very Important Remark about Notation (ORDER IS CRITICAL):

Notice that saying triangle ABC is congruent to triangle DEF is not the same as saying triangle ABC is congruent to triangle FED. For example the first statement means, among other things, that AB = DE and angle A = angle D. The second statement says that AB = FE and angle A = angle F. This is very different!

The notation convention for congruence subtly includes information about which vertices correspond. To write a correct congruence statement, the implied order must be the correct one.

The good feature of this convention is that if you tell me that triangle XYZ is congruent to triangle CBA, I know from the notation convention that XY = CB, angle X = angle C, etc. So once the order is set up properly at the beginning, it is easy to read off all 6 congruences.

Congruence Criteria

It turns out that knowing some of the six congruences of corresponding sides and angles are enough to guarantee congruence of the triangle and the truth of all six congruences.

Side-Angle-Side (SAS)

This criterion for triangle congruence is one of our axioms. So we do not prove it but use it to prove other criteria.

Using words:

If two sides in one triangle are congruent to two sides of a second triangle, and also if the included angles are congruent, then the triangles are congruent.

Using labels:

If in triangles ABC and DEF, AB = DE, AC = DF, and angle A = angle D, then triangle ABC is congruent to triangle DEF.

Side- Side-Side (SSS)

Using words:

If 3 sides in one triangle are congruent to 3 sides of a second triangle, then the triangles are congruent.

Using labels:

If in triangles ABC and DEF, AB = DE, BC = EF, and CA = FD, then triangle ABC is congruent to triangle DEF.

Proof: This was proved by using SAS to make "copies" of the two triangles side by side so that together they form a kite, including a diagonal. Then using what was proved about kites, diagonal cuts the kite into two congruent triangles.

Details of this proof are at this link. The similarity version of this proof is B&B Principle 8.

Step-by-step explanation:

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Answer 2

Given,

DG bisects  ∠EDF

DE=6, DF=8, EF=10

To find,

Find EG.

Solution,

We can solve this problem by the following method-

Given that DE, DF, EF are 6,8,10 respectively.

Then, DE²+DF²=EF².

Therefore, triangle DEF is a right-angled triangle.

Since EF is the longest side and its opposite angle is ∠D.

So, ∠D= 90°

DG bisects ∠D. Thus, we get two triangles.

ΔDGF and ΔDGE.

DG is the common side. So, ∠DFG= ∠DEG

∠FDG=∠EDG=45°.

Therefore, GE= FG (GE is opposite to ∠DEG and FG is opposite to ∠DFG )

If, EF=10

Then, GE=FG= 10 ÷ 2=

              = 5

Thus, EG is equal to 5.