First, look at the figure and see a few things:

You are told that the angle at F is perpendicular. Then ALL FOUR angles at F must be 90 degrees (and you now have four "right-angle triangles).

Next, we know that an angle places exactly on the circumference (like B and D) subtends an arc equal to double the angle. The total circle is 360 degrees, therefore "double" the sum of B+D = 360

The sum of B+D is half of that (180)

therefore D = 180 - B

This does not (yet) give us actual values, but it will help us later.

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22.

Note: in geometry, orientation is as important as value. The strange symbol ≅ means "congruent" and its meaning is: "angle ABD has the same value as angle CBD, even though it is not oriented the same way".

Geeks would say "angle ABD is congruent to CBD" (means the same thing)

The rest of question 22 is an algebra problem (not a geometry problem) where you have to write and equation, then solve for "x".

You are given

angle ABD = (2x/3)+10

angle CBD = (x/4) + 35

You have just been told (before that) the angles have the SAME value.

Therefore, these two things must be equal to each other

(2x/3) + 10 = (x/4) + 35

From then on, it is "just an algebra problem"

put each side on a common denominator (3 on the left, 4 on the right)

2x/3 + 30/3 = x/4 + 140/4

multiply both sides by 4 (gets rid of the denominator on the right)

8x/3 + 120/3 = x + 140

multiply both sides by 3 (gets rid of the denominator on the left)

8x + 120 = 3x + 420

subtract 120 from both sides (it "moves" the 120 to the right)

8x = 3x + 420 - 120

8x = 3x + 300

subtract 3x from both sides (it "moves" the 3x to the left)

8x - 3x = 300

5x = 300

divide both sides by 5

x = 60

Once you have a value for x, you can find the angles ABD and CBD

angle ABD = (2x/3)+10 = 2(60)/3 + 10 = 50

angle CBD = (x/4) + 35 = 60/4 + 35 = 50

(they are equal, as promised by the question)

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Before I even look at the other questions, I now realize that I know the value of ALL the angles in the figure.

In a triangle, the sum of angles is always 180.

I also remember that total D = 180 - total B = 180 - 100 = 80

(D is smaller than B, even though the drawing makes it look bigger)

(once we have real numbers, the numbers are more important than the figure)

Remember that all angles "F" are 90 degrees.

Triangle ABF: F=90 B=50, therefore A = 180 - 90 - 50 = 40

Triangle CBF: F=90 B=50, therefore C = 40

Triangle ABC: A=40, C=40, total B = 100 (= 50+50 as calculated in number 22)

Triangle AFD: F=90, D=40 (half of "total D" as calculated using total B), then A=50

Triangle CFD: F=90, D=40, therefore C=50

This makes "total A" = 40+50 = 90 and same thing for "total C" (angle BCD),

which makes sense: line BD is a diameter, therefore the arc covers 180 degrees on each side, and angles A and C each subtend an arc of 180 (therefore BAD = half of 180).

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23 begins by confirming what we have just found out: angles BAC and BCA have the same value (we know it is 40)

It then gives us another algebra problem to find the length of the sides.

The key is that: in triangle BAC, because we have two equal angles, therefore we have an isosceles triangle (two sides must be equal, and they are the sides facing the equal angles)

The given information is:

AB = 4x - 15

BC = 2x + 25

The two are euqal, we write

4x - 15 = 2x + 25

we "solve for x":

4x - 2x = 25 + 15

2x = 40

x = 20

We use this to find the sides

AB = 4x - 15 = 4(20) - 15 = 80-15 = 65 units

BC = 2x + 25 = 2(20) + 25 = 40+25 = 65

They are equal to each other (not surprising) and have a length of 65 units each.

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The rest of your questions are based on the same pattern: they are algebra problems where two things are equal, and you use this fact to build an equation, where you "solve for x".

In question 25, they confirm what we already know: sides AB and BC are equal.

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Careful:

In questions 26 and 27, they change the figure (even though they don't tell you directly).

They change the angles BAC and BCA to 60 degrees each (instead of 40).

This makes the triangle BAC an "equilateral triangle" (three equal angles at 60 each, three equal sides).

Once you understand their "trick", you quickly find the answers.

Just remember "equilateral" = equal sides AND equal angles.