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Basile [38]
3 years ago
9

Prove ΔPAB is isosceles.

Mathematics
1 answer:
Licemer1 [7]3 years ago
6 0

Answer:

See explanation

Step-by-step explanation:

If PX\cong PY, then triangle PXY is isosceles triangle. Angles adjacent to the base XY of an isosceles triangle PXY are congruent, so

\angle 1\cong \angle 2

and

m\angle 1=m\angle 2

Angles 1 and 3 are supplementary, so

m\angle 3=180^{\circ}-m\angle 1

Angles 2 and 4 are supplementary, so

m\angle 4=180^{\circ}-m\angle 2

By substitution property,

m\angle 4=180^{\circ}-m\angle 2=180^{\circ}-m\angle 1=m\angle 3

Hence,

\angle 3\cong \angle 4

Consider triangles APX and BPY. In these triangles:

  • PX\cong PY - given;
  • \angle 5\cong \angle 6 - given;
  • \angle 3\cong \angle 4 - proven,

so \triangle APX\cong \triangle BPY by ASA postulate.

Congruent triangles have congruent corresponding sides, then

AP\cong BP

Therefore, triangle APB is isosceles triangle (by definition).

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Answer:

The GCF for the variable part is  

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Step-by-step explanation:

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Steps to find the GCF for  

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,

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:

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2. Find the GCF for the variable part  

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3. Multiply the values together

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The factors for  

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1

,

2

,

3

,

6

,

9

,

18

The factors for  

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1

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3

,

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,

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1

,

3

,

5

,

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List all the factors for  

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,

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to find the common factors.

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:  

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2

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3

,

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,

9

,

18

15

:  

1

,

3

,

5

,

15

The common factors for  

18

,

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are  

1

,

3

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1

,

3

The GCF for the numerical part is  

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GCF

Numerical

=

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Next, find the common factors for the variable part:

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,

k

3

The factor for  

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1

is  

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The factors for  

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3

are  

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⋅

k

⋅

k

.

k

⋅

k

⋅

k

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3

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The common factor for the variables  

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Multiply the GCF of the numerical part  

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.

3

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