A single point does not fix the direction of a line, so many lines can pass through it. Two distinct points determine a unique straight line.
Rihan can draw infinitely many lines through one point. Sheetal can draw exactly one line through two given distinct points.
In the chain $L-M-P-Q-R$, each adjacent pair forms one segment. End points belong to one segment, while the middle joining points belong to two adjacent segments.
The line segments are $LM$, $MP$, $PQ$ and $QR$. Points $L$ and $R$ are on exactly one line segment. Points $M$, $P$ and $Q$ are on two line segments.
A ray is named by writing its starting point first. In ray $NB$, the starting point is $N$, not $T$.
The rays are $TA$, $TB$, $TN$ and $NB$. No, $T$ is the starting point of $TA$, $TB$ and $TN$, but not of $NB$.
Each drawing should show the stated incidence relation: meeting, intersecting, lying on a line, or lying away from a line.
a. Draw two line segments or rays $OP$ and $OQ$ with common point $O$. b. Draw lines $XY$ and $PQ$ crossing at $M$. c. Draw a line $l$ through $E$ and $F$, and mark $D$ away from the line. d. Draw segment or line $AB$ and mark $P$ on it.
The named points lie on the figure. A line can be named by any two points on it. Rays are named with their starting point first, and line segments are named by their two endpoints.
a. Five points: $D,E,O,B,C$. b. A line: $DE$ or $DB$. c. Four rays: $OC$, $OB$, $OE$, $OD$. d. Five line segments: $DE$, $DO$, $DB$, $EO$, $EB$.
The first letter in the name of a ray gives its starting point. Other points on the same ray may be used as the second letter.
a. Yes. It can also be named $OB$ because $O$ is the starting point and $B$ lies on the same ray. b. No. $OA$ starts at $O$, while $AO$ would start at $A$, so they are different rays.
An angle is formed by two rays with a common starting point. The common starting point is the vertex.
Yes. One possible angle is $\angle BDC$, with vertex $D$ and arms/rays $DB$ and $DC$.
Both arms have the same starting point $S$, so $S$ must be the middle letter when naming the angle.
Draw two rays $ST$ and $SR$ starting from the same point $S$. The angle is $\angle TSR$ or $\angle RST$, and its vertex is $S$.
When several rays meet at the same vertex, using only the vertex name is ambiguous. Three-letter notation identifies the exact arms of the angle.
$\angle APB$ cannot be labelled only as $\angle P$ because more than one angle is formed at point $P$ in the figure.
The common vertex is $T$. Use one point from each arm with the vertex in the middle.
The marked angles are $\angle RTQ$ and $\angle RTP$.
Three non-collinear points form a triangle. Each pair of points determines one line, and each point can be used as the vertex of one angle of the triangle.
You get three lines: $AB$, $BC$ and $CA$. You can name three angles: $\angle ABC$, $\angle BCA$ and $\angle CAB$; equivalently, $\angle CBA$, $\angle ACB$ and $\angle BAC$.
Four points with no three collinear determine one line for each pair of points: $\binom{4}{2}=6$. At each point, three rays to the other points form three angles, giving $4\times3=12$ named angles.
You get six lines: $AB$, $AC$, $AD$, $BC$, $BD$ and $CD$. The angles include $\angle BAC$, $\angle CAD$, $\angle BAD$, $\angle ADB$, $\angle BDC$, $\angle ADC$, $\angle DCA$, $\angle ACB$, $\angle DCB$, $\angle CBD$, $\angle DBA$ and $\angle CBA$.
Compare the angles by superimposition or by observing the amount of opening between the fold and the side of the paper.
A sample fold creates angles such as $\angle AEF$, $\angle BEF$, $\angle DFE$ and $\angle CFE$. In one such fold, $\angle AEF$ and $\angle CFE$ are larger than $\angle BEF$ and $\angle DFE$. The largest and smallest angles depend on the fold made.
$\angle AOB$ contains smaller parts including $\angle XOY$, so it is greater. In part c, the two angles are equal by symmetry in the given figure.
a. $\angle AOB$ is greater. b. $\angle AOB$ is greater. c. Neither is greater; $\angle XOB=\angle XOC$.
When angles are close in size or placed differently, visual comparison may be misleading. Folding, tracing or measuring can compare them accurately.
From the printed figure alone, it is not reliable to decide by sight. Superimposition or measurement is needed.
A straight angle is half a full turn and contains two right angles. Therefore one right angle is half of a straight angle, which is $\frac{1}{4}$ of a full turn.
A right angle is $\frac{1}{4}$ of a full turn.
A right angle looks like an exact L-shape and measures $90^\circ$.
A usual rectangular classroom window has four right angles. Other right angles may be seen at the corners of the blackboard, books, door frames, floor tiles and desks.
A straight angle is formed when the two arms are opposite rays of the same line. So the new point must be collinear with $A$ and $B$.
Join $A$ to grid points that lie on the same straight line through $A$ and $B$, on either side if available.
A right angle at $A$ is formed by a line through $A$ that divides the straight angle on line $AB$ into two equal parts.
Join $A$ to any grid point lying on a line through $A$ that is perpendicular to line $AB$.
Two perpendicular creases intersect to form four equal angles around a point. Since a full turn is $360^\circ$, each is $360^\circ\div4=90^\circ$, a right angle.
a. Four right angles are formed. b. Fold the paper so that the slanting crease falls exactly on itself in the opposite direction; the new crease made by the fold is perpendicular to the first crease.
Use the definitions: acute $<90^\circ$, right $=90^\circ$, obtuse between $90^\circ$ and $180^\circ$, and straight $=180^\circ$.
Angles smaller than a right angle are acute; angles exactly like an L-shape are right; angles greater than a right angle but less than a straight angle are obtuse; angles forming a straight line are straight.
Draw any angles less than $90^\circ$ for acute angles and any angles greater than $90^\circ$ but less than $180^\circ$ for obtuse angles.
Examples of acute angles: $30^\circ$, $45^\circ$, $60^\circ$. Examples of obtuse angles: $110^\circ$, $125^\circ$, $150^\circ$.
The names describe the visual appearance of the angle openings.
The word acute fits because an acute angle has a small, sharp opening. The word obtuse fits because an obtuse angle has a wider, blunter opening.
The number of acute angles increases by 9 each time: $3,12,21,30,\ldots$. Thus the next count is $21+9=30$.
The numbers are 3, 12 and 21. The next figure will have 30 acute angles.
Read the protractor marks from the common vertex $A$ and count the degree units between the arms of each angle.
$\angle KAL=30^\circ$, $\angle WAL=50^\circ$, and $\angle TAK=120^\circ$.
Each angle is measured at the common vertex $O$ by subtracting the protractor positions of its two arms.
$\angle POQ=35^\circ$, $\angle POR=95^\circ$, $\angle POS=125^\circ$, $\angle POT=160^\circ$, $\angle QOR=60^\circ$, $\angle QOS=90^\circ$, $\angle QOT=125^\circ$, $\angle QOU=145^\circ$, $\angle ROS=30^\circ$, $\angle ROT=65^\circ$, $\angle ROU=85^\circ$, $\angle SOT=35^\circ$, $\angle SOU=55^\circ$, $\angle TOU=20^\circ$.
The straight angle of $180^\circ$ is divided into 8 equal parts. Therefore each part measures $180^\circ\div8=22.5^\circ$.
Each angle is $22.5^\circ$.
Place the protractor centre at the vertex and read the degree mark where the second arm meets the scale.
The angle measures are approximately $47^\circ$, $23^\circ$ and $108^\circ$ for the three given diagrams.
Use a protractor to measure real classroom angles by placing its centre on the vertex and aligning one arm with the baseline.
Sample observations: a book corner is $90^\circ$, a door opened partly may be about $60^\circ$, and a clock-hands angle at 3 o'clock is $90^\circ$.
A standard protractor is needed when the angle placement or available space does not suit the handmade paper protractor.
The angle measures are $42^\circ$ and $116^\circ$. A paper protractor cannot be used conveniently here.
Measure the smaller unmarked angle as $100^\circ$. The full turn is $360^\circ$, so the marked reflex angle is $360^\circ-100^\circ=260^\circ$.
The marked angle is $260^\circ$.
Measure each angle with a protractor, keeping the centre at the vertex and one arm along the baseline.
a. $80^\circ$ b. $120^\circ$ c. $60^\circ$ d. $130^\circ$ e. $130^\circ$ f. $60^\circ$
Read each ray's position on the protractor and subtract the relevant degree marks to find the angle between the rays.
$\angle BXE=115^\circ$, $\angle CXE=85^\circ$, $\angle AXB=65^\circ$ and $\angle BXC=30^\circ$.
All three angles have vertex $Q$ and arm $QP$ as one side. Measure from $QP$ to $QR$, $QS$ and $QT$ respectively.
$\angle PQR=45^\circ$, $\angle PQS=100^\circ$ and $\angle PQT=150^\circ$.
List angles by choosing a vertex where two segments or rays meet, then name one point on each arm with the vertex in the middle. Measures should be found with a protractor.
Some angles possible are $\angle CAP$, $\angle ACD$, $\angle APL$, $\angle DLP$, $\angle RPL$, $\angle SLP$, $\angle PRS$, $\angle LSR$, $\angle BRS$ and $\angle CLP$.
Each hour step corresponds to $30^\circ$. Multiply $30^\circ$ by the number of hour steps between the hands.
a. The clock face is divided into 12 equal parts, so each part is $360^\circ\div12=30^\circ$. b. At 2 o'clock the angle is $60^\circ$, at 4 o'clock it is $120^\circ$, and at 6 o'clock it is $180^\circ$. c. For example, at 3 o'clock the smaller angle is $90^\circ$.
The amount of opening is the rotation of the door about the hinge.
Yes. The vertex is at the hinge where the door meets the wall. The arms are the edge or position of the door and the wall or closed-door position.
The swing moves by rotation about its support. The starting angle measures how far the rope has been turned from its resting position.
The angle can be seen between the resting vertical position of the swing rope and the pulled-back position of the rope before release.
A slope can be described by the angle made between the slanting surface and a reference line.
Yes. Angles can describe the slopes of the slabs. One arm is along the slanting slab and the other arm can be taken as the horizontal or vertical reference direction. The slanting slab arm is visible; the reference arm may be imagined and may not be visible in the toy.
Use the degree-measure definitions of acute, obtuse and reflex angles to choose suitable rays from $A$.
Draw from $A$ to a grid point making an angle less than $90^\circ$ for an acute angle, between $90^\circ$ and $180^\circ$ for an obtuse angle, and greater than $180^\circ$ but less than $360^\circ$ for a reflex angle.
Classify using the measure: acute is less than $90^\circ$, obtuse is between $90^\circ$ and $180^\circ$, and reflex is between $180^\circ$ and $360^\circ$.
a. $\angle PTR=30^\circ$ (acute). b. $\angle PTQ=60^\circ$ (acute). c. $\angle PTW=102^\circ$ (obtuse). d. $\angle WTP=258^\circ$ (reflex).
Any drawing is acceptable if it contains exactly the required types of angles: three less than $90^\circ$, one equal to $90^\circ$, and two between $90^\circ$ and $180^\circ$.
One valid figure can have angles $40^\circ$, $50^\circ$, $60^\circ$ as the three acute angles, $90^\circ$ as the right angle, and $110^\circ$, $120^\circ$ as the two obtuse angles.
Use a protractor to mark the two side turns as $40^\circ$ and the central turn as $60^\circ$, then join the arms to form the letter M.
Draw an M whose left and right side angles are each $40^\circ$ and whose middle angle is $60^\circ$.
The three angles around the joining point must add to $360^\circ$: $150^\circ+60^\circ+150^\circ=360^\circ$.
Draw a Y with the upper two arms separated by $60^\circ$, and each upper arm making $150^\circ$ with the lower stem on the outside.
A full turn is $360^\circ$. With 24 equally spaced spokes, adjacent spokes make $360^\circ\div24=15^\circ$. Acute multiples are $15^\circ,30^\circ,45^\circ,60^\circ,75^\circ$; the next, $90^\circ$, is right, so the largest acute angle is $75^\circ$.
The angle between two neighbouring spokes is $15^\circ$. The largest acute angle formed between two spokes is $75^\circ$.
Let the angle be $x^\circ$. Since $4x$ is acute, $4x<90$, so $x<22.5$. Since $5x$ is obtuse, $90<5x<180$, so $x>18$. Whole-number values satisfying both are $19,20,21,22$.
For whole-number degree measures, the possibilities are $19^\circ$, $20^\circ$, $21^\circ$ and $22^\circ$.
$\angle REB$ is a straight angle, so it measures $180^\circ$. Since $\angle TER=80^\circ$, $\angle BET=180^\circ-80^\circ=100^\circ$. Also, $\angle BES=90^\circ$, so $\angle SET=100^\circ-90^\circ=10^\circ$.
$\angle BET=100^\circ$ and $\angle SET=10^\circ$.