Quiz #1 For MAPEH Math 10a

An angle inscribed in a semicircle is a right angle because the diameter of the semicircle forms a straight line, and any angle formed by a straight line is always equal to 180 degrees. Since a semicircle is exactly half of a circle, the angle formed by the diameter is 180/2 = 90 degrees, which is a right angle.

If two inscribed angles intercept the same arc or congruent arcs, it means that they are formed by two intersecting chords or tangents that intersect at the same point on the circle. Since the intercepted arcs are congruent, it implies that the chords or tangents are also congruent. Therefore, the angles formed by these chords or tangents will also be congruent. Hence, the statement is true.

If a line is tangent to a circle, it means that it touches the circle at only one point. In this case, if we draw a radius from the center of the circle to the point of tangency, the line will be perpendicular to this radius. This is because a tangent line forms a right angle with the radius at the point of tangency. Therefore, the statement is true.

Two chords of a circle are congruent if they are equidistant from the center. This means that the distance from the center of the circle to each chord is the same. Congruent chords have the same length and are parallel to each other.

In the given figure, segment AB is equal to 2x - 19 and segment CD is equal to x + 1. To find the value of x, we can set these two expressions equal to each other and solve for x. So, 2x - 19 = x + 1. By simplifying the equation, we get x = 20. Therefore, the value of x is 20.

In the given figure, we are given that arc AB is equal to 9x + 3 degrees and arc CD is equal to 83 - x degrees. We need to find the value of x. By equating the two expressions, we can set up the equation 9x + 3 = 83 - x. Solving this equation, we get 10x = 80, which implies x = 8. Therefore, the value of x is 8.

Tangent circles are circles that touch each other at exactly one point, known as the point of tangency. In this case, the circles are also tangent to a common line at the same point. Therefore, the correct answer is "tangent circles".

A common tangent is a line that is tangent to each of two circles, but not necessarily noncoplanar circles. Therefore, the given statement is false.

If two segments from the same external point are tangent, it means they touch the same circle at that point. Tangent segments that touch the same circle from an external point are always equal in length. Therefore, the correct answer is congruent.

The statement is false. The measure of an angle formed by two tangents to the same circle is equal to one-half the positive difference of the measures of the intercepted arcs, not the negative difference.

Since CD is congruent to BC, it means that they have the same length. Therefore, CD = BC. Given that CD is 7x - 10 and BC = 39, we can set up an equation: 7x - 10 = 39. Solving this equation, we add 10 to both sides and get 7x = 49. Finally, we divide both sides by 7, which gives us x = 7.

When two circles are internally tangent, it means that one circle is contained entirely within the other circle. In this case, the centers of both circles, the point of tangency (where the circles touch), and the radii connecting the centers and the point of tangency all lie on the same line. This line is called the line of centers, and it is collinear, meaning all the points lie on the same straight line. Therefore, the correct answer is collinear.

The measurement of angle BAC can be determined by subtracting the measure of arc DE from the measure of arc BC. Since arc DE is 179 degrees and arc BC is 59 degrees, the difference between the two is 120 degrees. Therefore, angle BAC measures 120 degrees.

If two arcs of a circle are inscribed between two parallel secants, they are not necessarily complementary. The term "complementary" means that the sum of the two angles formed by the arcs would be 90 degrees. However, in this case, the angles formed by the arcs can have any value depending on the specific positions of the secants. Therefore, the statement is false.

In the given figure, angle ACB is the angle formed between the lines AC and CB. The angles given in the figure are irrelevant to finding angle ACB. Therefore, we can conclude that angle ACB is 62 degrees.

To find angle AOC in degrees, we need to use the property that angles in the same segment of a circle are equal. Since angle AOC is an angle in the same segment as angle ABC, and angle ABC is given as 104 degrees, we can conclude that angle AOC is also 104 degrees.

The measurement of angle ADC in degrees is 52 because angle ADC is an inscribed angle that intercepts the same arc as angle ABC. According to the inscribed angle theorem, the measure of an inscribed angle is half the measure of its intercepted arc. Since angle ABC intercepts a 104-degree arc, angle ADC will have half of that measure, which is 52 degrees.

In the given figure, angle ECB is given as 49 degrees. The measure of a major arc is always twice the measure of the corresponding central angle. So, the measure of major arc CB can be found by multiplying the measure of angle ECB by 2, which gives us 98 degrees. However, since the given options do not include 98 degrees, we can conclude that the measure of major arc CB is 262 degrees.

A tangent is a line that intersects a circle at exactly one point. It touches the circle at that one point and does not pass through the circle. Therefore, the correct answer is "TANGENT, TANGENT LINE, TANGENT LINES".

Opposite angles of an inscribed quadrilateral are supplementary. This means that the sum of the measures of two opposite angles in an inscribed quadrilateral is always 180 degrees. Therefore, the correct answer is "SUPPLEMENTARY, SUPPLEMENTARY ANGLES."