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leva [86]
3 years ago
6

Match the statements with their values

Mathematics
1 answer:
Elan Coil [88]3 years ago
8 0
Tile 1:
In any triangle (regardless its type), the sum of measures of the internal angles is 180°.
This means that:
∠ABC + ∠BAC +  ∠ACB = 180°

Tile 2:
The sum of measures of internal angles of a triangle is 180°.
We are given that:
ΔABC is isosceles where AB = AC 
This means that:
∠ABC = ∠ACB
We are also given that measure angle BAC is 70 degrees
180 = ∠ABC + ∠ACB + 70
∠ABC + ∠ACB = 110°
We know that both angles are equal, therefore:
∠ABC = ∠ACB = 110/2 = 55°

Tile 3:
We are given that ΔQPR is an isosceles triangle where PQ = QR
This means that:∠QPR = ∠QRP
We are given that ∠QRP = 30°
This means that:∠QPR = 30°

Tile 4:
A diagram representing the given scenario is attached.
Now we have:
point D is midpoint to AB and point E is midpoint to BC
There is a theorem stating that: "In a triangle, a line joining the midpoints of two sides is parallel to the third side and equals half its length"
Applying this to the givens, we would conclude that:ED is parallel to AC
Now, since these two lines are parallel, then angles BAC and BDE are corresponding angles which means that they are equal.
This means that:∠BAC = ∠BDE = 45°

Hope this helps :)

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The variable x represents the number of angelfish Carlos bought and the variable y represents the number of parrotfish he bought
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Segments AB , CD , and EF intersect at point O, points A, E, C and points B, F, D are collinear so that AO ≅ OB , CO ≅ OD . Prov
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If points A, E and C are colinear, then they lie on the same line. The same statement you can say about points B, F and D.

1. Consider triangles AOC and BOD. In these triangles:

  • AO≅OB (given);
  • CO≅OD (given);
  • ∠AOC≅∠BOD (as vertical angles).

Thus, ΔAOC≅ΔBOD by SAS Postulate (If any two corresponding sides and their included angle are the same in both triangles, then the triangles are congruent). Corresponding parts of congruent triangles are congruent, then

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Since angles ACO and BDO are alternate interior angles between lines AE and BF with transversal CD and these angles are congruent, then lines AE and BF are parallel.

This gives you that

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  • ∠ECO≅∠ODF.

2. Consider triangles ECO and FDO. In these triangles

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  • ∠ECO≅∠ODF (previous proof).

Therefore, ΔECO≅ΔFDO by AAS Postulate (if two angles and the non-included side one triangle are congruent to two angles and the non-included side of another triangle, then these two triangles are congruent). Then CE≅FD.

3. Note that

  • AE=AC+CE;
  • BF=BD+DF.

Since AC≅BD and CE≅DF, then AE=AC+CE=BD+DF=BF.

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