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## Chengde Feng – Angles and Areas – Recap

Chengde Feng’s lecture was greeted with very high attendance and very active participation.  Throughout his lecture, Mr. Feng showed the students how to solve a wide variety of geometry problems involving area.  These are problems that frequently occur in contests like MATHCOUNTS or AMC.  Many shortcuts and techniques were shared to help shave off precious sections during these problem solving tests.

You knew you were at a Math Circle when in the second hour, Mr. Feng announced a “pop quiz” and instead of groans there was honest enthusiasm from the students eager to apply their new knowledge.

Mr. Feng has provided the problems from the lecture. Members of the Metroplex Math Circle e-mail group can download these files from the group site. To join the e-mail group simply click below. Click to join MetroplexMathCircle

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## Chengde Feng – Angles and Areas – November 1

Chengde Feng, an instructor at AwesomeMath, father of Zuming Feng and a good friend of the Metroplex Math Circle will come to Dallas this Saturday to lecture on “Angles and Areas.”  To help us prepare, Mr. Feng has sent along this list of facts that every Middle School Student should know.

### Concepts You Need to Know

1.  Angles of a Triangle

• Vertical angles are congruent
• The sum of the measures of the angles of a triangle is 180.
• The measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles.

2. Angles of a Polygon

• The sum of the measures of the interior angles of a n-gon is ${(n-2)180}$
• The measure of each interior angle of a regular n-gon is $\dfrac{(n-2)180}{n}$
• The sum of the measures of the exterior angles of any convex polygon, one angle at each vertex, is 360.

3. Triangles

• The length of each side of a triangle must be less than the sum of the lengths of the other two sides.
• If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
• An equilateral triangle has three $60^\circ$ angles.
• Pythagorean Theorem In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.

4.  Areas The area of:

• a rectangle equals the product of its base and height.
• a parallelogram equals the product of its base and height.
• a triangle equals half the product of its base and height.
• an equilateral triangle with side length l is $\dfrac{\sqrt{3}}{4}t^2$
• a rhombus equals half of the product of its diagonals.
• a trapezoid equals half the product of the height and the sum of the bases.
• a circle with radius r is $\pi r^2$ and its circumference is $2 \pi r$.
• a sector AOB of circle O with radius r is $\dfrac{m\angle AOB}{360}\pi r^2$

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