This page is made for the postulates and theorems of Unit 5. By: Sai Govindu.
Polygon Angle Sum Theorem - The sum of the interior angle measures of a convex polygon with n sides is (n – 2)180°. It is helpful to calculate not only the sum obviously but also 1 interior angle and since we know an interior angle and it's exterior angle are supplementary, we can know 1 exterior angle in a convex polygon as well so this theorem is very useful.
- If ABC is a convex triangle, it’s interior sum=180 because of Polygon Angle Sum Theorem:(3-2)180=180
- If DEFGH is a convex pentagon, it’s interior sum=540 because of P.A.S.T. : (5-2)180=540
Polygon Exterior Angle Sum Theorem- The sum of the exterior angle measures, one angle at each vertex, of a convex polygon is 360°. – always true This theorem can help you also figure out the measure of 1 exterior angle which can be useful when calculating other units as well.
- If this is a convex triangle, then the exterior angles add up to 360
Parallelogram Property Theorems(proving) - if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram; this is a property that can prove if a quadrilateral is a parallelogram by drawing and measuring the bisector length and midpoint if on a coordinate graph; if both pairs of sides and/or angles are congruent(opposites), then the quadrilateral is a parallelogram
Segments CE and EB are congruent(12.45 cm) while segments AE and ED(8.8 cm) are congruent. This proves that each of the segments bisect each other as E is the midpoint and according to the property, we know that this is a parallelogram. Also, angles CDE and CAB are congruent while angles DCA and DBA are also congruent. In addition, sides CD and AB are congruent while sides CA and DB are also congruent. This also proves that this quadrilateral is a parallelogram. To add, if you notice, the consecutive angles of the quadrilateral are up to exactly 180 degrees which is known as a supplementary angle pair. If at least one angle has 2 consecutive angle relationships, then we can prove or know that this quadrilateral is a parallelogram. There are many ways to prove if a quadrilateral is a parallelogram as shown here.
Rectangle Theorems(ways to prove) - If a quadrilateral is a parallelogram and the diagonals inside are also congruent(we already know they bisect), then we know that the parallelogram is a rectangle. This theorem is very useful when determining if a parallelogram is a rectangle and is a useful property of rectangles used in proofs. We can also conclude that if at least one of the angles in the parallelogram is a right angle, it's a rectangle.
In rectangle ABCD, the diagonals AC and BD are congruent to each other which tells us this is a rectangle because the diagonals in a rectangle are always congruent. The 4 corner angles are 90 degrees or right angles which also is a theorem or way to prove that this is a rectangle. m<BAD = 90 degrees
Rhombus Theorems(ways to prove) - If a quadrilateral is a parallelogram and all 4 sides are congruent, then the quadrilateral is a rhombus. This theorem is probably the most used theorem when proving if a quadrilateral is a rhombus. If a quadrilateral is a parallelogram and the diagonals in the interior are perpendicular bisectors of each other, then the quadrilateral is a rhombus. This theorem is useful because we know that all the angles created from the diagonals are 90 degree angles and we can use this for solving other parts of a rhombus. If a quadrilateral is a parallelogram and the diagonals bisect each pair of their corresponding opposite angles, then the quadrilateral is a rhombus. This theorem is also important because when trying to prove or solve for the rhombus angles, we know if the diagonals are present, that they bisect the angle and each angle on different sides of the diagonal are congruent.
Sides AB, BC, CD, and DA all measure up to 8,81 cm which means they're congruent. The measure of angle AFD is 90 degrees and if solved for, all the interior angles with F as the center vertex are 90 degrees telling us that these diagonals AC and BD are perpendicular bisectors(already know they are bisector because of parallelogram properties). Lastly, angles ADF and CDF are congruent like angles ABD and DBC telling us that diagonals AC and BD bisect each corresponding pair of opposite angles.
Square Theorems(ways to prove) - If a quadrilateral is a parallelogram and it's a rectangle and rhombus(has all their properties), then the quadrilateral is also a square. This theorem is significant because if a shape is a rhombus and rectangle, it's a square and this can be used when proving by first proving a quadrilateral a rectangle and then a rhombus and then using this 1 and only theorem to then prove the quadrilateral is indeed a square.
The measure of angle DCA is 90 degrees so we already know this is a rectangle. All 4 sides are congruent so we already know this is also a rhombus. With these 2 components, we can conclude that this is a square.
Significance of these Postulates and Theorems : I chose the angle sum theorem because it's a universal rule for all polygons in case you need to find one of the angles of a convex or the sum for other proofs. I chose exterior angle sum theorem because when doing proofs, you know that all the exterior angles add up to 360 so you can find one exterior angle and then 1 interior angle because an exterior angle and its corresponding interior angle are supplementary angles. I chose all of the parallelogram property theorems because you have to use and know each of these theorems when proving in parallelogram proofs. I chose the theorems that deal with identifying or determining if a parallelogram is a rectangle because it's significant to know when proving special parallelograms and when going on in the quadrilateral family with rhombuses and squares.