Geometry Chapter 4 Test (4.1 To 4.6)

1.

I f space y o u space a r e space g i v e n space t h a t space increment space S U N space approximately equal to space increment space T A N comma space y o u space c a n space c o n c l u d e space t h a t :

Angle S is congruent to Angle A

Segment SN is congruent to Segment TN

Angle T is congruent to Angle U

Segment SU is congruent to Segment TN

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Explanation

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2. $T h e space f i g u r e space t o space t h e space r i g h t space s h o w s space increment space A B C approximately equal to space _ _ _ _ _ _$

Triangle CDE

Triangle DEC

Triangle ECD

Triangle EDC

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Explanation

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3. The measures of the angles of a triangle are 40°, 45°, and 95°. The triangle is:

Obtuse

Right

Acute

Equilateral

The sum of the angles of a triangle is always 180 degrees. In this case, the sum of the given angles (40° + 45° + 95°) is 180°, which means it is a valid triangle. Since one of the angles (95°) is greater than 90°, the triangle is classified as obtuse.

Explanation

The sum of the angles of a triangle is always 180 degrees. In this case, the sum of the given angles (40° + 45° + 95°) is 180°, which means it is a valid triangle. Since one of the angles (95°) is greater than 90°, the triangle is classified as obtuse.

4.

W r i t e space a space t w o minus c o l u m n space p r o o f. space M a r k space y o u r space d i a g r a m. space space space G i v e n : space L space i s space t h e space m i d p o i n t space o f space top enclose J M end enclose. space space top enclose J K end enclose space approximately equal to top enclose N M end enclose space space space space space P r o v e : space increment J K L approximately equal to increment M N L space space

L is the midpoint of JM

Segment JK is congruent to Segment NM

Angle J is congruent to Angle M

Triangle JKL is congruent to Triangle MNL

Submit

6. If you are told that C is the midpoint of

top enclose A D end enclose

and

top enclose E B end enclose

which postulates and theorems would you need to use in order to prove that

top enclose A B end enclose space parallel to top enclose E D end enclose

Mark the diagram!

SAS, CPCTC, and then if alternate interior angles are congruent, lines are parallel

ASA, CPCTC, and then if corresponding angles are congruent, lines are parallel

SSS, CPCTC, and then if alternate interior angles are congruent, lines are parallel

HL, CPCTC, and then if corresponding angles are congruent, lines are parallel

The correct answer is SAS, CPCTC, and then if alternate interior angles are congruent, lines are parallel. This is because the given information states that C is the midpoint of AB, which implies that AC=CB. This satisfies the SAS (Side-Angle-Side) criterion for congruence. CPCTC (Corresponding Parts of Congruent Triangles are Congruent) can then be used to show that corresponding angles in the congruent triangles are congruent. Finally, if alternate interior angles are congruent, it implies that the lines are parallel.

Explanation

The correct answer is SAS, CPCTC, and then if alternate interior angles are congruent, lines are parallel. This is because the given information states that C is the midpoint of AB, which implies that AC=CB. This satisfies the SAS (Side-Angle-Side) criterion for congruence. CPCTC (Corresponding Parts of Congruent Triangles are Congruent) can then be used to show that corresponding angles in the congruent triangles are congruent. Finally, if alternate interior angles are congruent, it implies that the lines are parallel.

Submit

10. State the postulate or theorem that proves the two triangles are congruent. (Mark the diagram) $top enclose W Y end enclose space approximately equal to space top enclose X Y end enclose comma space top enclose W Z end enclose space approximately equal to top enclose X Z end enclose$

SSS

SAS

ASA

AAS

The correct answer is SSS. According to the Side-Side-Side (SSS) postulate, if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent. In the given diagram, the three sides of the triangles are congruent, which satisfies the SSS postulate and proves the triangles are congruent.

Explanation

The correct answer is SSS. According to the Side-Side-Side (SSS) postulate, if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent. In the given diagram, the three sides of the triangles are congruent, which satisfies the SSS postulate and proves the triangles are congruent.

11. $W r i t e space a space t w o minus c o l u m n space p r o o f. space M a r k space y o u r space d i a g r a m. space space space space G i v e n : space top enclose E G end enclose space approximately equal to space top enclose I A end enclose space space space space space space space space space space space space space space angle E G A space approximately equal to space angle I A G space space space space P r o v e : space angle G E A space approximately equal to space angle A I G space space space$

Segment EG is congruent to Segment IA

Angle EGA is congruent to Angle IAG

Segment GA is congruent to Segment GA

Triangle EGA is congruent to Triangle IAG

Angle GEA is congruent to Triangle AIG

Submit

13. $Q u a d r i l a t e r a l space M N Q P space i s space m a d e space o f space t w o space c o n g r u e n t space t r i a n g l e s. space stack N P with bar on top space b i s e c t s space angle M N Q space a n d space angle M P Q. space space I n space t h e space q u a d r i l a t e r a l comma space m angle M N Q space equals space 50 space a n d space m angle M P Q space equals space 100. space space W h a t space i s space t h e space m e a s u r e space o f space angle M space ?$

25

50

60

105

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Explanation

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14. Use the diagram to solve for x.

27

64

38

40

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Explanation

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15. State the postulate or theorem that proves the two triangles are congruent. (Mark the diagram)

space angle P space a n d space angle R space a r e space r i g h t space a n g l e s ; space space space top enclose Q T end enclose space approximately equal to space top enclose S T end enclose

SSS

SAS

ASA

AAS

The postulate or theorem that proves the two triangles are congruent is the AAS (Angle-Angle-Side) postulate. According to this postulate, if two angles and the non-included side of one triangle are congruent to the corresponding angles and non-included side of another triangle, then the two triangles are congruent. In the given diagram, if we have two angles in one triangle that are congruent to two angles in another triangle, and the non-included side between these angles is also congruent, then we can conclude that the triangles are congruent by the AAS postulate.

Explanation

The postulate or theorem that proves the two triangles are congruent is the AAS (Angle-Angle-Side) postulate. According to this postulate, if two angles and the non-included side of one triangle are congruent to the corresponding angles and non-included side of another triangle, then the two triangles are congruent. In the given diagram, if we have two angles in one triangle that are congruent to two angles in another triangle, and the non-included side between these angles is also congruent, then we can conclude that the triangles are congruent by the AAS postulate.