Volumes & Surfaces Areas : Volumes & Surfaces Areas 3
Slide 2: 2 2
Slide 4: Surface Area of Any Prism
(b is the shape of the ends)
Surface Area = Lateral area + Area of two ends
(Lateral area) = (perimeter of shape b) * L
Surface Area = (perimeter of shape b) * L+ 2*(Area of shape b)
RHS Congruence : RHS Congruence RHS-Right Angle Hypotenuse Side!
When the right angle and the hypotenuse and the given side are equal for a right angle triangle then we say that the given 2 triangles are congruent.
Example proving RHS congruence : Example proving RHS congruence E D A F C B <B=<E=90 degrees
AC=DF (hypotenuse)
BC=EF (given side)
Example proving AAA congruence : Example proving AAA congruence A B P O Q In this figure QA and PB are perpendiculars to AB. If AO is equal to 10cm, BO equal to 6cm, & PB equal to 9cm, Find AQ. Let us consider the triangles OAQ and OBP congruent.
<A=<B A
<AOQ=<BOP (vertically opposite angles) A
<P=<Q (corresponding) A 10 AQ
6 = 9
90=6AQ
AQ=15
In a parallelogram if one angle A is equal to 110 degrees find the remaining angles? : In a parallelogram if one angle A is equal to 110 degrees find the remaining angles? All sides of a parallelogram have to equal 360 degrees. So if Angle A is 110 degrees then
360=110 + B + C + D
-110=- 110
250= B + C + D
D also =s 110
360-220= 140
So B & C = 70.
If the diagonals of a parallelogram are equal, then show it is a rectangle? : If the diagonals of a parallelogram are equal, then show it is a rectangle? Theorem 11.1 If ABCD is a parallelogram then its nonconsecutive sides and its nonconsecutive angles are equal.
Proof We need to prove that AB = CD, BC = AD.
SASSide/Angle/Side : SASSide/Angle/Side SAS- If 2 sides and the included angle are congruent to 2 sides and the included angle of a 2nd triangle, the 2 triangles are congruent. And included angle is an angle created by 2 sides of a triangle.
SSSSide/Side/Side : SSSSide/Side/Side It is a rule that is used in geometry to prove triangles congruent. The rule states that if 3 sides on 1 triangle are congruent to 3 sides of a 2nd triangle, the 2 triangles are congruent.
AAAAngle/Angle/Angle : AAAAngle/Angle/Angle If in 2 triangles, corresponding angles are equal, then their corresponding sides are in the same ratio and hence the 2 triangles are similar. b a f c d e <a=<d
<b=<e
<c=<f
ASAAngle/Side/Angle : ASAAngle/Side/Angle ASA is a rule used in geometry to prove triangles are congruent. The rule states that if 2 angles and the included side of one triangle are congruent to 2 angles and the included side of another triangle, the triangles are congruent.
AASAngle/Angle/Side : AASAngle/Angle/Side AAS is used in geometry to prove triangles are congruent. The rules state that if 2 angles and a non-included side of 1 triangle are congruent to 2 angles and the corresponding non-included side of another triangle the 2 triangles are congruent.
CPCTCCorresponding Parts of Congruent Triangle Are Congruent/Equal : CPCTCCorresponding Parts of Congruent Triangle Are Congruent/Equal When 2 triangles are congruent, all 6 pairs of corresponding parts {angles & sides} are congruent. This statement is usually simplified as corresponding parts of congruent triangles are congruent. If
then the following conditions are true:
E q u i v a l e n c er e l a t i o n s : E q u i v a l e n c er e l a t i o n s Reflexivity: a ~ a
*Every triangle is congruent to itself
Symmetry: if a ~ b then b ~ a
Transitive: if a ~ b and b ~ c then a ~ c.
Slide 20: 1 2 3 4 5 6 7 8 <1, <5
<2,<6
<3,<7
<4,<8 Corresponding angles <3, <5
<4, <6 Alternate Interior Angles <1, <7
<2, <8 Alternate Exterior Angles In geometry, adjacent angles are angles that have a common ray coming out of the vertex going between two other rays. Ex. Of adjacent Angles
Supplementary Angles : Supplementary Angles A pair of angles are supplementary if their respective measures sum to 180°.
If the two supplementary angles are adjacent (i.e. have a common vertex and share a side, but do not have any interior points in common) their non-shared sides form a line.
Complementary Angles : Complementary Angles A pair of angles are complementary if the sum of their angles is 90°.
If the two complementary angles are adjacent (i.e. have a common vertex and share a side, but do not have any interior points in common) their non-shared sides form a right angle.
Slide 23: Area of Circle= πr²
Arc length= circumference-2πr * Area/360
Arc length=Circumference multiplied by πr² divided by 2πr
Area of a sector= A= mAB/360 * πr² (
What π = : What π =
Slide 26: l- length
b- base
h- height
W- width
a- just a side
s- side