Geometry basics: points, lines and angles  (Page 3/3)

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Once angles can be measured, they can then be compared. For example, all right angles are 90 ${}^{\circ }$ , therefore all right angles are equal and an obtuse angle will always be larger than an acute angle.

The following video summarizes what you have learnt so far about angles.

Note that for high school trigonometry you will be using degrees, not radians as stated in the video.

Special angle pairs

In [link] , straight lines $AB$ and $CD$ intersect at point X, forming four angles: $\stackrel{^}{{X}_{1}}$ or $\angle BXD$ , $\stackrel{^}{{X}_{2}}$ or $\angle BXC$ , $\stackrel{^}{{X}_{3}}$ or $\angle CXA$ and $\stackrel{^}{{X}_{4}}$ or $\angle AXD$ .

The table summarises the special angle pairs that result.

 Special Angle Property Example adjacent angles share a common vertex and a common side $\left(\stackrel{^}{{X}_{1}},\stackrel{^}{{X}_{2}}\right)$ , $\left(\stackrel{^}{{X}_{2}},\stackrel{^}{{X}_{3}}\right)$ , $\left(\stackrel{^}{{X}_{3}},\stackrel{^}{{X}_{4}}\right)$ , $\left(\stackrel{^}{{X}_{4}},\stackrel{^}{{X}_{1}}\right)$ linear pair (adjacent angles on a straight line) adjacent angles formed by two intersecting straight lines that by definition add to 180 ${}^{\circ }$ $\stackrel{^}{{X}_{1}}+\stackrel{^}{{X}_{2}}={180}^{\circ }$ ; $\stackrel{^}{{X}_{2}}+\stackrel{^}{{X}_{3}}={180}^{\circ }$ ; $\stackrel{^}{{X}_{3}}+\stackrel{^}{{X}_{4}}={180}^{\circ }$ ; $\stackrel{^}{{X}_{4}}+\stackrel{^}{{X}_{1}}={180}^{\circ }$ opposite angles angles formed by two intersecting straight lines that share a vertex but do not share any sides $\stackrel{^}{{X}_{1}}=\stackrel{^}{{X}_{3}}$ ; $\stackrel{^}{{X}_{2}}=\stackrel{^}{{X}_{4}}$ supplementary angles two angles whose sum is 180 ${}^{\circ }$ complementary angles two angles whose sum is 90 ${}^{\circ }$
The opposite angles formed by the intersection of two straight lines are equal. Adjacent angles on a straight line are supplementary.

The following video summarises what you have learnt so far

Parallel lines intersected by transversal lines

Two lines intersect if they cross each other at a point. For example, at a traffic intersection two or more streets intersect; the middle of the intersection is the common point between the streets.

Parallel lines are lines that never intersect. For example the tracks of a railway line are parallel (for convenience, sometimes mathematicians say they intersect at 'a point at infinity', i.e. an infinite distance away). We wouldn't want the tracks to intersect after as that would be catastrophic for the train!

All these lines are parallel to each other. Notice the pair of arrow symbols for parallel.

Interesting fact

A section of the Australian National Railways Trans-Australian line is perhaps one of the longest pairs of man-made parallel lines.

Longest Railroad Straight (Source: www.guinnessworldrecords.com) The Australian National Railways Trans-Australian line over the Nullarbor Plain, is 478 km (297 miles) dead straight, from Mile 496, between Nurina and Loongana, Western Australia, to Mile 793, between Ooldea and Watson, South Australia.

A transversal of two or more lines is a line that intersects these lines. For example in [link] , $AB$ and $CD$ are two parallel lines and $EF$ is a transversal. We say $AB\parallel CD$ . The properties of the angles formed by these intersecting lines are summarised in the table below.

 Name of angle Definition Examples Notes interior angles the angles that lie inside the parallel lines in [link] $a$ , $b$ , $c$ and $d$ are interior angles the word interior means inside adjacent angles the angles share a common vertex point and line in [link] ( $a$ , $h$ ) are adjacent and so are ( $h$ , $g$ ); ( $g$ , $b$ ); ( $b$ , $a$ ) exterior angles the angles that lie outside the parallel lines in [link] $e$ , $f$ , $g$ and $h$ are exterior angles the word exterior means outside alternate interior angles the interior angles that lie on opposite sides of the transversal in [link] ( $a,c$ ) and ( $b$ , $d$ ) are pairs of alternate interior angles, $a=c$ , $b=d$ co-interior angles on the same side co-interior angles that lie on the same side of the transversal in [link] ( $a$ , $d$ ) and ( $b$ , $c$ ) are interior angles on the same side. $a+d={180}^{\circ }$ , $b+c={180}^{\circ }$ corresponding angles the angles on the same side of the transversal and the same side of the parallel lines in [link] $\left(a,e\right)$ , $\left(b,f\right)$ , $\left(c,g\right)$ and $\left(d,h\right)$ are pairs of corresponding angles. $a=e$ , $b=f$ , $c=g$ , $d=h$

The following video summarises what you have learnt so far

Euclid's Parallel line postulate. If a straight line falling across two other straight lines makes the two interior angles on the same side less than two right angles (180 ${}^{\circ }$ ), the two straight lines, if produced indefinitely, will meet on that side. This postulate can be used to prove many identities about the angles formed when two parallel lines are cut by a transversal.
1. If two parallel lines are intersected by a transversal, the sum of the co-interior angles on the same side of the transversal is 180 ${}^{\circ }$ .
2. If two parallel lines are intersected by a transversal, the alternate interior angles are equal.
3. If two parallel lines are intersected by a transversal, the corresponding angles are equal.
4. If two lines are intersected by a transversal such that any pair of co-interior angles on the same side is supplementary, then the two lines are parallel.
5. If two lines are intersected by a transversal such that a pair of alternate interior angles are equal, then the lines are parallel.
6. If two lines are intersected by a transversal such that a pair of alternate corresponding angles are equal, then the lines are parallel.

Find all the unknown angles in the following figure:

1. $\text{AB}\parallel \text{CD}$ . So $x={30}^{°}$ (alternate interior angles)
2. $\begin{array}{ccc}\hfill 160+y& =& 180\hfill \\ \hfill y& =& {20}^{°}\hfill \end{array}$
(co-interior angles on the same side)

Determine if there are any parallel lines in the following figure:

1. Line EF cannot be parallel to either AB or CD since it cuts both these lines. Lines AB and CD may be parallel.
2. We can show that two lines are parallel if we can find one of the pairs of special angles. We know that ${\stackrel{ˆ}{E}}_{2}={25}^{°}$ (opposite angles). And then we note that
$\begin{array}{ccc}\hfill {\stackrel{ˆ}{E}}_{2}& =& {\stackrel{ˆ}{F}}_{4}\hfill \\ & =& {25}^{°}\hfill \end{array}$
So we have shown that $\text{AB}\parallel \text{CD}$ (corresponding angles)

Angles

1. Use adjacent, corresponding, co-interior and alternate angles to fill in all the angles labeled with letters in the diagram below:
2. Find all the unknown angles in the figure below:
3. Find the value of $x$ in the figure below:
4. Determine whether there are pairs of parallel lines in the following figures.
5. If AB is parallel to CD and AB is parallel to EF, prove that CD is parallel to EF:

The following video shows some problems with their solutions

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Virgil
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CYNTHIA
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Harper
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s.
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SUYASH
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Ebrahim
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s.
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tahir
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Cied
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Stotaw
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Azam
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Prasenjit
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Prasenjit
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Azam
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Uday
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Uday
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