## In the figure below a b c and d are collinear fc is parallel

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## The graph of a certain hyperbola, y = h(x), is shown in the standard (x,y) coordinate plane below.

APB forms a straight line in the diagram above. What is the measure, in degrees, of angle CPB if the measure of angle APC is eighty-one degrees greater than the measure of angle DPB and the measurements of angles CPD and DPB are equal?
Reason for this: Let x be the angle DPB’s length. We may represent the measure of angle APC as x + 81 since it is eighty-one degrees greater than the measure of angle DPB. Also, since the measure of angle CPD is equal to the measure of angle DPB, we can represent the measure of CPD as x.
This implies that the angles DPB and CPD have the same measure of 33 degrees. The original question asks us to find the angle CPB’s measure, which is equal to the number of the angles DPB and CPD’s measures.
We can solve for y in terms of x by subtracting x from both sides since x + y = 180. In other terms, y = 180 – x. We can then solve for x by substituting this value into the equation (1/2)y = 2x.
Angle ABC is 36 degrees in length. The original query, on the other hand, asks us to find the measure of ABC’s complement, which we previously denoted as z. Since the sum of an angle’s and its complement’s measures equals 90, we may write the following equation:

### In the figure below, points a and b are on opposite banks of a small stream

Angles, like regular numbers, can be applied to produce a count, which can be used to determine the measure of an unknown angle. We may also figure out what angle is missing since we know the number has to be a certain value. Remember — the number of the degree scales of angles in any triangle equals 180 degrees. The triangle ABC is depicted below, with angle A equaling 60 degrees, angle B equaling 50 degrees, and the sum of all three angles equaling 180 degrees. So, the measure of angle A + angle B + angle C = 180 degrees. This is valid for every triangle in the field of geometry. This concept can be used to determine the measure of an angle(s) when the degree measure is missing or unavailable.
You don’t have to solve the equation every time you plug in certain values. If you’ve had the hang of it, you’ll be able to say things like “okay, 40 + 60 = 100, so the other angle has to be 80!” It’s also a lot faster.
The smallest angle in the given ratio is defined by the smallest number. Isn’t 4 the smallest number given? Since this is a ratio, we must multiply all of the numbers (4,5,9) by a common factor to obtain the actual angles. (For example, with a factor of 20, 60 and 80 are in a 3:4 ratio.)

### Ms. hernandez began her math class by saying

Thales’ theorem states that the angle ABC is a right angle if A, B, and C are distinct points on a circle and the line AC is a diameter. Thales’s theorem is a special case of the inscribed angle theorem and is stated and proven as part of the 31st proposition in the third book of Euclid’s Elements. [1] It is usually attributed to Thales of Miletus, but Pythagoras has also been mentioned.
Before Thales proved it, Indian and Babylonian mathematicians knew it for special cases.
[4] During his travels to Babylon, Thales is said to have discovered that an angle inscribed in a semicircle is a right angle.
[5] The theorem is named after Thales because he was said by ancient sources to have been the first to prove the theorem, using his own results that the base angles of an isosceles triangle are identical, and that the number of angles in a triangle is equal to 180°.
displaystyle beginaligned&m ABcdot m BC&m ABcdot m BC&m ABcdot m BC&m ABcdot m BC&m ABcdot m BC&m_
[8pt]=&frac sin theta cos theta +1cdot frac sin theta cos theta -1cdot frac sin theta cos theta -1
[8pt]={}&{\frac {\sin ^{2}\theta }{\cos ^{2}\theta -1}}\\
[8pt]=&frac sin 2theta -sin 2theta