Honors Geometry - Polygon Angle-Sum Theorems, Kites and Trapezoids, Properties of Parallelograms, Rectangle + Rhombus = Square

Okay, here are some problems based on the lessons:

1. Polygon Angle-Sum Theorem:

   • What is the interior angle sum of a 20-gon?

   • How many sides does a polygon have if its interior angles measure 160^{\circ}?

2. Polygon Exterior Angle-Sum Theorem:

   • The exterior angles of a pentagon measure (x + 10)^{\circ}, (2x)^{\circ}, (x + 20)^{\circ}, (3x - 10)^{\circ}, and (3x - 20)^{\circ}. Find the value of x and the measure of each angle.

3. Kites and Trapezoids:

   • In kite ABCD, AB = AD and CB = CD. If m\angle BAC = 35^{\circ} and m\angle BCD = 44^{\circ}, find m\angle ABC.

   • In isosceles trapezoid EFGH, EF \parallel GH. If m\angle E = 110^{\circ}, find m\angle G.

   • The bases of a trapezoid are 8 inches and 12 inches. Find the length of the midsegment.

4. Properties of Parallelograms:

   • In parallelogram PQRS, m\angle P = (5x - 10)^{\circ} and m\angle Q = (3x + 20)^{\circ}. Find the measures of angles P and Q.

   • The diagonals of parallelogram WXYZ intersect at point A. If WA = 3y + 2 and YA = 5y - 8, find the length of diagonal WY.

5. Rectangle, Rhombus, and Square:

   • The diagonals of rectangle ABCD intersect at point E. If AE = 2x + 5 and DE = 3x - 1, find the length of AE.

   • In rhombus FGHI, m\angle FGI = 52^{\circ}. Find m\angle FHI.

   • Determine whether the points A(1, 2), B(4, 2)$

To solve the problems, you'll need to apply the properties and theorems associated with each shape. Here's a general approach:

1. Polygon Angle-Sum Theorem: Use the formulas for interior and exterior angles of polygons.

   • Interior Angle Sum: (n-2) \times 180^{\circ}, where n is the number of sides.

   • Each Interior Angle (for a regular polygon): ((n-2) \times 180^{\circ}) / n

   • Exterior Angle Sum: Always 360^{\circ}

2. Kites and Trapezoids: Use the properties of kites (e.g., diagonals are perpendicular) and trapezoids (e.g., midsegment is the average of the bases).

3. Parallelograms: Use the properties of parallelograms (e.g., opposite angles are congruent, diagonals bisect each other).

4. Rectangle, Rhombus, and Square: Use the properties specific to each shape (e.g., rectangles have congruent diagonals, rhombi have perpendicular diagonals, squares have all properties of both rectangles and rhombi).

Okay, here are some problems based on the lessons: 1.
Polygon Angle-Sum Theorem:

• What is the interior angle sum of a 20-gon?

• How many sides does a polygon have if its interior angles measure 160^{\circ}?

1. Polygon Exterior Angle-Sum Theorem:

• The exterior angles of a pentagon measure (x + 10)^{\circ}, (2x)^{\circ}, (x + 20)^{\circ}, (3x - 10)^{\circ}, and (3x - 20)^{\circ}. Find the value of x and the measure of each angle.

1. Kites and Trapezoids:

• In kite ABCD, AB = AD and CB = CD. If m\angle BAC = 35^{\circ} and m\angle BCD = 44^{\circ}, find m\angle ABC.

• In isosceles trapezoid EFGH, EF \parallel GH. If m\angle E = 110^{\circ}, find m\angle G.

• The bases of a trapezoid are 8 inches and 12 inches. Find the length of the midsegment.

1. Properties of Parallelograms:

• In parallelogram PQRS, m\angle P = (5x - 10)^{\circ} and m\angle Q = (3x + 20)^{\circ}. Find the measures of angles P and Q.

• The diagonals of parallelogram WXYZ intersect at point A. If WA = 3y + 2 and YA = 5y - 8, find the length of diagonal WY.

1. Rectangle, Rhombus, and Square:

• The diagonals of rectangle ABCD intersect at point E. If AE = 2x + 5 and DE = 3x - 1, find the length of AE.

• In rhombus FGHI, m\angle FGI = 52^{\circ}. Find m\angle FHI.

• Determine whether the points A(1, 2), B(4, 2)$ To solve the problems, you'll need to apply the properties and theorems associated with each shape. Here's a general approach:

1. Polygon Angle-Sum Theorem: Use the formulas for interior and exterior angles of polygons.

• Interior Angle Sum: (n-2) \times 180^{\circ}, where n is the number of sides.

• Each Interior Angle (for a regular polygon): ((n-2) \times 180^{\circ}) / n

• Exterior Angle Sum: Always 360^{\circ}

1. **Kites