farid zen

1. A polygon has a side of length 385 .

A similar polygon has a corresponding side of length 77 .
How do the areas of the two polygons compare to each other?

1. Suppose that a triangle has base z and altitude x . The sides of the triangle are scaled by a factor of 7 to get a similar triangle.

What is a formula for the area of the new triangle?

1. Suppose that a polygon has perimeter 36 .

The sides of the polygon are scaled by a factor of 7 to get a similar polygon.
What is the perimeter of the scaled polygon? Suppose that the sides of a polygon are scaled by a factor of 4 to get a similar polygon.
This new polygon has perimeter 116 .
What is the perimeter of the original polygon?

1. Suppose that the sides of a polygon are scaled by a factor of 4 to get a similar polygon.

This new polygon has perimeter 116 .
What is the perimeter of the original polygon?

If two sides (CA and CB) and the included angle ( BCA ) of a triangle are congruent to the corresponding two sides (C'A' and C'B') and the included angle (B'C'A') in another triangle, then the two triangles are congruent.



Example 1: Let ABCD be a parallelogram and AC be one of its diagonals. What can you say about triangles ABC and CDA? Explain your answer.

Solution to Example 1:

• In a parallelogram, opposite sides are congruent. Hence sides
  
  BC and AD are congruent, and also sides AB and CD are congruent.
  
  
• In a parallelogram opposite angles are congruent. Hence angles
  
  ABC and CDA are congruent.
  
  
• Two sides and an included angle of triangle ABC are congruent to two corresponding sides and an included angle in triangle CDA. According to the above postulate the two triangles ABC and CDA are congruent.
  
  

Side-Side-Side (SSS) Congruence Postulate

If the three sides (AB, BC and CA) of a triangle are congruent to the corresponding three sides (A'B', B'C' and C'A') in another triangle, then the two triangles are congruent.



Example 2: Let ABCD be a square and AC be one of its diagonals. What can you say about triangles ABC and CDA? Explain your answer.

Solution to Example 2:

• In a square, all four sides are congruent. Hence sides
  
  AB and CD are congruent, and also sides BC and DA are congruent.
  
  
• The two triangles also have a common side: AC. Triangles ABC has three sides congruent to the corresponding three sides in triangle CDA. According to the above postulate the two triangles are congruent. The triangles are also right triangles and isosceles.
  
  

Angle-Side-Angle (ASA) Congruence Postulate

If two angles (ACB, ABC) and the included side (BC) of a triangle are congruent to the corresponding two angles (A'C'B', A'B'C') and included side (B'C') in another triangle, then the two triangles are congruent.



Example 3: ABC is an isosceles triangle. BB' is the angle bisector. Show that triangles ABB' and CBB' are congruent.

Solution to Example 3:

• Since ABC is an isosceles triangle its sides AB and BC are congruent and also its angles BAB' and BCB' are congruent. Since BB' is an angle bisector, angles ABB' and CBB' are congruent.
  
  Two angles and an included side in triangles ABB' are congruent to two corresponding angles and one included side in triangle CBB'. According to the above postulate triangles ABB' and CBB' are congruent.

Angle-Angle-Side (AAS) Congruence Theorem

If two angles (BAC, ACB) and a side opposite one of these two angles (AB) of a triangle are congruent to the corresponding two angles (B'A'C', A'C'B') and side (A'B') in another triangle, then the two triangles are congruent.



Example 4: What can you say about triangles ABC and QPR shown below.

Solution to Example 4:

• In triangle ABC, the third angle ABC may be calculated using the theorem that the sum of all three angles in a triangle is equal to 180 derees. Hence
  
  angle ABC = 180 - (25 + 125) = 30 degrees
  
  
• The two triangles have two congruent corresponding angles and one congruent side.
  
  angles ABC and QPR are congruent. Also angles BAC and PQR are congruent. Sides BC and PR are congruent.
  
  
• Two angles and one side in triangle ABC are congruent to two corresponding angles and one side in triangle PQR. According to the above theorem they are congruent.

Right Triangle Congruence Theorem

If the hypotenuse (BC) and a leg (BA) of a right triangle are congruent to the corresponding hypotenuse (B'C') and leg (B'A') in another right triangle, then the two triangles are congruent.



Example 5: Show that the two right triangles shown below are congruent.

Solution to Example 5:

• We first use Pythagora's theorem to find the length of side AB in triangle ABC.
  
  length of AB = sqrt [5 ² - 3 ²] = 4
  
  
• One leg and the hypotenuse in triangle ABC are congruent to a corresponding leg and hypotenuse in the right triangle A'B'C'. According to the above theorem, triangles ABC and B'A'C' are congruent.

1. Problem: Is triangle PQR congruent to
           triangle STV by SAS? Explain.


 Solution: Segment PQ is congruent
             to segment ST because
             PQ = ST = 4.
           Angle Q is congruent to
             angle T because
             angle Q = angle T = 100 degrees.
           Segment QR is congruent
             to segment TV because QR = TV = 5.
           Triangle PQR is congruent
             to triangle STV by Side-Angle-Side.

Back to Top

---

Side-Side-Side is a rule used in geometry to prove triangles congruent. The rule states that if three sides of one triangle are congruent to three sides of a second triangle, the two triangles are congruent.

1. Problem: Show that triangle QYN is congruent
           to triangle QYP.


 Solution: Segment QN is congruent to
             segment QP and segment YN is
             congruent to segment YP because that
             information is given in the figure.
           Segment YQ is congruent to segment
             YQ by the Reflexive Property of Con-
             gruence, which says any figure is
             congruent to itself.
           Triangle QYN is congruent to triangle
             QYP by Side-Side-Side.

Back to Top

---

Angle-Side-Angle is a rule used in geometry to prove triangles are congruent. The rule states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent. An included side is a side that is common to (between) two angles. For example, in the figure used in the problem below, segment AB is an included side to angles A and B.

1.   Problem: Show that triangle BAP is congruent
             to triangle CDP.







    Solution: Angle A is congruent to angle D
                 because they are both right angles.
              Segment AP is congruent to segment DP be-
                 cause both have measures of 5.
              Angle BPA and angle CPD are congruent be-
                 cause vertical angles are congruent.
              Triangle BAP is congruent to triangle CDP
                 by Angle-Side-Angle.

Back to Top

---

Angle-Angle-Side is a rule used in geometry to prove triangles are congruent. The rule states that if two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of another triangle, the two triangles are congruent.

1.   Problem: Show that triangle CAB is congruent
             to triangle ZXY.







    Solution: Angle A and angle Y are congruent
                 because that information is given
                 in the figure.
              Angle C is congruent to angle Z
                 because that information is given
                 in the figure.
              Segment AB corresponds to segment XY and
                 they are congruent because that
                 information is given in the figure.
              Triangle CAB is congruent to triangle ZXY
                 by Angle-Angle-Side.

Back to Top

---

When two triangles are congruent, all six pairs of corresponding parts (angles and sides) are congruent. This statement is usually simplified as corresponding parts of congruent triangles are congruent, or CPCTC for short.

1.   Problem: Prove segment BC is congruent to
             segment CE.
              



 



    Solution: First, you have to prove that triangle
              CAB is congruent to triangle CED.
             
              Angle A is congruent to angle D
                 because that information is given
                 in the figure.
              Segment AC is congruent to segment CD
                 because that information is given
                 in the figure.
              Angle BCA is congruent to angle DCE
                 because vertical angles are
                 congruent.
              Triangle CAB is congruent to triangle CED
                 by Angle-Side-Angle.
             
             
              Now that you know the triangles are
              congruent, you know that all
              corresponding parts must be congruent. 
              By CPCTC, segment BC
              is congruent to segment CE.

1. Problem: Find the value of x, y, and
           the measure of angle P.



 Solution: To find the value of x and y,
           write proportions involving corresponding
           sides.  Then use cross products to solve.
          
           4   x          4   7
           - = -          - = -
           6   9          6   y
          
           6x = 36        4y = 42
          
           x = 6          y = 10.5
          
           To find angle P, note that angle P
           and angle S are corresponding angles.
           By definition of similar polygons,
           angle P = angle S = 86o.

---

The triangle, geometry's pet shape :-) , has a couple of special rules dealing with similarity. They are outlined below.

1. Angle-Angle Similarity - If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

1. Problem: Prove triangle ABE is similar
           to triangle CDE.


 Solution: Angle A and angle C are congruent (this
           information is given in the figure).
          
           Angle AEB and angle CED are
           congruent because vertical angles are
           congruent.
          
           Triangle ABE and triangle CDE are similar
           by Angle-Angle.

2. Side-Side-Side Similarity - If all pairs of corresponding sides of two triangles are proportional, then the triangles are similar.

3. Side-Angle-Side Similarity - If one angle of a triangle is congruent to one angle of another triangle and the sides that include those angles are proportional, then the two triangles are similar.

2. Problem: Are the triangles shown in
           the figure similar?


 Solution: Find the ratios of the
           corresponding sides.
          
           UV    9   3           VW   15   3
           -- = -- = -           -- = -- = -
           KL   12   4           LM   20   4
          
           The sides that include angle V
           and angle L are proportional.
          
           Angle V and angle L are
           congruent (the information is given in
           the figure).
          
           Triangle UVS and triangle KLM
           are similar by Side-Angle-Side.
1.   Problem: Find PT and PR







    Solution: 4   x
              - = --      because the sides are divided
              7   12      proportionally when you draw a
                          parallel line to another side.
                         
             
              7x = 48     Cross products
             
             
              x = 48/7
             
             
              PT = 48/7
              PR = 12 + 48/7 = 132/7