Geometry Quick Review: Angles Formed By Transversals (Quick Review Notes)
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Parallel Lines - Angles Formed by Transversals Reference/Graphic Organizer
Save my name, email, and website in this browser for the next time I comment. It usually denotes the direction of line 5 Collinear points — If three or more points lie on the same line, they are called collinear points, otherwise they are called non-collinear points. If a transversal intersects two parallel lines, then i each pair of corresponding angles is equal. If a transversal interacts two lines such that, either i any one pair of corresponding angles is equal, or ii any one pair of alternate interior angles is equal or iii any one pair of interior angles on the same side of the transversal is supplementary ,then the lines are parallel.
Lines which are parallel to a given line are parallel to each other. If two parallel lines are intersected by a transversal, the bisectors of any pair of alternate interior angles are parallel and vice-versa. Real Number Properties:. Reflexive Property. If equal quantities are added to equal quantities, the sums are equal.
If equal quantities are subtracted from equal quantities, the differences are equal. If equal quantities are multiplied by equal quantities, the products are equal. If equal quantities are divided by equal nonzero quantities, the quotients are equal. A quantity may be substituted for its equal in any expression.
Ruler Postulate. Points on a line can be paired with the real numbers. Segment Addition Postulate. The whole is equal to the sum of its parts. The midpoint of a segment is a point on the segment forming two congruent segments equal segments. The bisector of a segment is a line, a ray, or segment which cuts the given segment into two congruent segments equal segments.
A straight line segment can be drawn joining any two points.
Parallel and Perpendicular Slopes lines
Angle Addition Postulate. Right Angles Euclid's Postulate 4. All right angles are congruent equal in measure. All straight angles are congruent equal in measure. Vertical angles are congruent equal in measure.
Summary - Angles
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles. Base Angle Theorem Isosceles Triangle. If two sides of a triangle are congruent, the angles opposite these sides are congruent equal in measure. Base Angle Converse Isosceles Triangle. If two angles of a triangle are congruent, the sides opposite these angles are congruent equal in length.
Pythagorean Theorem. The sum of the lengths of any two sides of a triangle must be greater than the third side. If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
Let AB be a chord of a circle not passing through its centre O. The chord and the two equal radii OA and BO form an isosceles triangle whose base is the chord. The angle AOB is called the angle at the centre subtended by the chord. In the module, Rhombuses, Kites and Trapezia we discussed the axis of symmetry of an isosceles triangle.
Translating that result into the language of circles:. Let AB be a chord of a circle with centre O. The following three lines coincide:. Constructions with radii and chords give plenty of opportunity for using trigonometry.
Fun Ideas for Special Angle Pairs
Show that:. This exercise shows that sine can be regarded as the length of the semichord AM in a circle of radius 1, and cosine as the perpendicular distance of the chord from the centre. Let A and B be two different points on a circle with centre O. These two points divide the circle into two opposite arcs. If the chord AB is a diameter, then the two arcs are called semicircles. Now join the radii OA and OB.
The two radii divide the circle into two sectors, called correspondingly the major sector OAB and the minor sector OAB. It is no surprise that equal chords and equal arcs both subtend equal angles at the centre of a fixed circle. The result for chords can be proven using congruent triangles, but congruent triangles cannot be used for arcs because they are not straight lines, so we need to identify the transformation involved.
A chord AB of a circle divides the circle into two segments. Otherwise, the two segments are called a major segment and a minor segment. Let AB be a diameter of a circle with centre O , and let P be any other point on the circle. This famous theorem is traditionally ascribed to the Greek mathematician Thales, the first known Greek mathematician. A quadrilateral whose diagonals are equal and bisect each other is a rectangle.
Explain why APB is a right angle. The angle in a semicircle theorem has a straightforward converse that is best expressed as a property of a right-angled triangle:. The circle whose diameter is the hypotenuse of a right-angled triangle passes through all three vertices of the triangle. Complete the rectangle ACBR. Because ACBR is a rectangle, its diagonals bisect each other and are equal. A set of points in the plane is often called a locus. The term is used particularly when the set of points is the curve traced out by a moving point.
For example, a circle can be defined as the locus of a point that moves so that its distance from some fixed point is constant. The two examples below use the converse of the angle in a semicircle theorem to describe a locus. A photographer is photographing the ornamental front of a building.
He wants the two ends of the front to subtend a right angle at his camera. Describe the set of all positions where he can stand. A plank of length metres is initially resting flush against a wall, but it slips outwards, with its top sliding down the wall and its foot moving at right angles to the wall.
What path does the midpoint of the plank trace out? Angles at the centre and circumference.
The angle-in-a-semicircle theorem can be generalised considerably. In each diagram below, AB is an arc of a circle with centre O , and P is a point on the opposite arc. We shall show that this relationship holds also for the other two cases, when the arc is a minor arc left-hand diagram or a major arc right-hand diagram.
An angle at the circumference of a circle is half the angle at the centre subtended by the same arc. Let AB be an arc of a circle with centre O, and let P be any point on the opposite arc. The proof divides into three cases, depending on whether:. A punter stands on the edge of a circular racing track. With his binoculars he is following a horse that is galloping around the track at one revolution a minute. Hence the punter is rotating his binoculars at a constant rate that is half the rate at which the horse is rotating about the centre.
This corollary of the previous theorem is a particularly significant result about angles in circles:. Some alternative terminology. The last two theorems are often expressed in slightly different language, and some explanation is needed to avoid confusion. An altitude of a triangle is a perpendicular from any of the three vertices to the opposite side, produced if necessary. The two cases are illustrated in the diagrams below. There are three altitudes in a triangle.
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The following theorem proves that they concurrent at a point called the orthocentre H of the triangle. It is surprising that circles can be used to prove the concurrence of the altitudes. The altitudes of a triangle are concurrent. In the module, Congruence , we showed how to draw the circumcircle through the vertices of any triangle. To do this, we showed that the perpendicular bisectors of its three sides are concurrent, and that their intersection, called the circumcentre of the triangle, is equidistant from each vertex. No other circle passes through these three vertices.
If we tried to take as centre a point P other than the circumcentre, then P would not lie on one of the perpendicular bisectors, so it could not be equidistant from the three vertices. When there are four points, we can always draw a circle through any three of them provided they are not collinear , but only in special cases will that circle pass through the fourth point. A cyclic quadrilateral is a quadrilateral whose vertices all lie on a circle.
This is the last type of special quadrilateral that we shall consider. Suppose that we are given a quadrilateral that is known to be cyclic, but whose circumcentre is not shown perhaps it has been rubbed out. The circumcentre of the quadrilateral is the circumcentre of the triangle formed by any three of its vertices, so the construction to the right will find its circumcentre. The distinctive property of a cyclic quadrilateral is that its opposite angles are supplementary.
The following proof uses the theorem that an angle at the circumference is half the angle at the centre standing on the same arc. Join the radii OB and OD. Here is an alternative proof using the fact that two angles in the same segment are equal.
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An exterior angle of a cyclic quadrilateral is supplementary to the adjacent interior angle, so is equal to the opposite interior angle. This gives us the corollary to the cyclic quadrilateral theorem:.
An exterior angle of a cyclic quadrilateral is equal to the opposite interior angle. This exterior angle and A are both supplementary to BCD , so they are equal. Show that AP CR in the diagram to the right. If a cyclic trapezium is not a rectangle, show that the other two sides are not parallel, but have equal length. The property of a cyclic quadrilateral proven earlier, that its opposite angles are supplementary, is also a test for a quadrilateral to be cyclic. That is the converse is true. If the opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic.
Construct the circle through A , B and D , and suppose, by way of contradiction, that the circle does not pass through C. If an exterior angle of a quadrilateral equals the opposite interior angle, then the quadrilateral is cyclic. In the diagram to the right, the two adjacent acute angles of the trapezium are equal. Prove that the trapezium is cyclic. The sine rule states that for any triangle ABC , the ratio of any side over the sine of its opposite angle is a constant,. Each term is the ratio of a length over a pure number, so their common value seems to be a length.
Thus it reasonable to ask, what is this common length? The proof of this result provides a proof of the sine rule that is independent of the proof given in the module, Further Trigonometry. It is sufficient to prove that is the diameter of the circumcircle. A tangent to a circle is a line that meets the circle at just one point. The diagram below shows that given a line and a circle, can arise three possibilities:. The point where a tangent touches a circle is called a point of contact. It is not immediately obvious how to draw a tangent at a particular point on a circle, or even whether there may be more than one tangent at that point.
Let T be a point on a circle with centre O. First we prove parts a and c. Let be the line through T perpendicular to the radius OT. Let P be any other point on , and join the interval OP. Hence P lies outside the circle, and not on it. This proves that the line is a tangent, because it meets the circle only at T. It also proves that every point on , except for T , lies outside the circle.